How many ways can 200 identical balls be distributed into 40 distinct jars?Distinct balls into distinct boxes with a minimal number of balls in each boxNumber of ways of choosing identical ballsHow many ways are there to place 7 distinct balls into 3 distinct boxes?Ways to distribute $400$ non distinct balls to $3$ bins such that in all of the bins there are more than $200$ balls or less than $100$ balls$12$ Identical balls can be placed into $3$ identical boxes,Total possible ways to arrange 40 identical balls in 3 different boxesChoosing distinct balls to put into indentical urns.How many ways are there to distribute 26 identical balls into six distinct boxes such that…Counting distributions of distinct balls into distinct boxes without PIEDistributing $x$ identical balls into $p$ distinct boxes, what is the probability that there are at most $n$ balls in any of the boxes?
Is it possible to use .desktop files to open local pdf files on specific pages with a browser?
How to align and center standalone amsmath equations?
Can I use my Chinese passport to enter China after I acquired another citizenship?
Proof of Lemma: Every nonzero integer can be written as a product of primes
Greatest common substring
Can the Supreme Court overturn an impeachment?
Could the E-bike drivetrain wear down till needing replacement after 400 km?
Longest common substring in linear time
Freedom of speech and where it applies
Can a significant change in incentives void an employment contract?
Is possible to search in vim history?
What does the Rambam mean when he says that the planets have souls?
My friend sent me a screenshot of a transaction hash, but when I search for it I find divergent data. What happened?
Confusion on Parallelogram
Engineer refusing to file/disclose patents
What linear sensor for a keyboard?
Visiting the UK as unmarried couple
A social experiment. What is the worst that can happen?
Can somebody explain Brexit in a few child-proof sentences?
Fly on a jet pack vs fly with a jet pack?
We have a love-hate relationship
Greco-Roman egalitarianism
Customize circled numbers
How do I repair my stair bannister?
How many ways can 200 identical balls be distributed into 40 distinct jars?
Distinct balls into distinct boxes with a minimal number of balls in each boxNumber of ways of choosing identical ballsHow many ways are there to place 7 distinct balls into 3 distinct boxes?Ways to distribute $400$ non distinct balls to $3$ bins such that in all of the bins there are more than $200$ balls or less than $100$ balls$12$ Identical balls can be placed into $3$ identical boxes,Total possible ways to arrange 40 identical balls in 3 different boxesChoosing distinct balls to put into indentical urns.How many ways are there to distribute 26 identical balls into six distinct boxes such that…Counting distributions of distinct balls into distinct boxes without PIEDistributing $x$ identical balls into $p$ distinct boxes, what is the probability that there are at most $n$ balls in any of the boxes?
$begingroup$
How many ways can $200$ identical balls be distributed into $40$ distinct jars so that number Balls in the first $20$ jars is greater than the number of balls in the last $20$ jars?
I came up with two different solutions to the problem but the answers come out differently. The first solution uses symmetry:
There are $binom23940$ possible ways to place $200$ balls in $40$ jars. We then subtract all possibilities where the number of balls in the first $20$ jars is equal to the number of balls in the last $20$ jars, which is equal to $binom11920^2$. We then exploit symmetry (the number of possibilities that there are more balls in the first $20$ jars is equal to the number of possibilities that there are more balls in the last twenty jars). The final answer is
$$fracbinom23940 - binom11920^22$$
The second solution uses a sum. It comes out to
$$sum_k=101^200 binomk+1920binom219-k20$$
combinatorics
$endgroup$
|
show 2 more comments
$begingroup$
How many ways can $200$ identical balls be distributed into $40$ distinct jars so that number Balls in the first $20$ jars is greater than the number of balls in the last $20$ jars?
I came up with two different solutions to the problem but the answers come out differently. The first solution uses symmetry:
There are $binom23940$ possible ways to place $200$ balls in $40$ jars. We then subtract all possibilities where the number of balls in the first $20$ jars is equal to the number of balls in the last $20$ jars, which is equal to $binom11920^2$. We then exploit symmetry (the number of possibilities that there are more balls in the first $20$ jars is equal to the number of possibilities that there are more balls in the last twenty jars). The final answer is
$$fracbinom23940 - binom11920^22$$
The second solution uses a sum. It comes out to
$$sum_k=101^200 binomk+1920binom219-k20$$
combinatorics
$endgroup$
1
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Mar 16 at 8:51
$begingroup$
Hi, welcome to MSE! Could you explain us, please, what do you mean by "($X$ choose $Y$)"?
$endgroup$
– Yanior Weg
Mar 16 at 8:52
$begingroup$
@YaniorWeg $C^x_y$ that is I think he means combination.
$endgroup$
– Bijayan Ray
Mar 16 at 8:57
$begingroup$
Is not the number of possible ways to place 200 balls in 40 jars $binom23939$?
$endgroup$
– user
Mar 16 at 10:03
$begingroup$
@user Still waking up. Yes, you are correct.
$endgroup$
– N. F. Taussig
Mar 16 at 10:26
|
show 2 more comments
$begingroup$
How many ways can $200$ identical balls be distributed into $40$ distinct jars so that number Balls in the first $20$ jars is greater than the number of balls in the last $20$ jars?
I came up with two different solutions to the problem but the answers come out differently. The first solution uses symmetry:
There are $binom23940$ possible ways to place $200$ balls in $40$ jars. We then subtract all possibilities where the number of balls in the first $20$ jars is equal to the number of balls in the last $20$ jars, which is equal to $binom11920^2$. We then exploit symmetry (the number of possibilities that there are more balls in the first $20$ jars is equal to the number of possibilities that there are more balls in the last twenty jars). The final answer is
$$fracbinom23940 - binom11920^22$$
The second solution uses a sum. It comes out to
$$sum_k=101^200 binomk+1920binom219-k20$$
combinatorics
$endgroup$
How many ways can $200$ identical balls be distributed into $40$ distinct jars so that number Balls in the first $20$ jars is greater than the number of balls in the last $20$ jars?
I came up with two different solutions to the problem but the answers come out differently. The first solution uses symmetry:
There are $binom23940$ possible ways to place $200$ balls in $40$ jars. We then subtract all possibilities where the number of balls in the first $20$ jars is equal to the number of balls in the last $20$ jars, which is equal to $binom11920^2$. We then exploit symmetry (the number of possibilities that there are more balls in the first $20$ jars is equal to the number of possibilities that there are more balls in the last twenty jars). The final answer is
$$fracbinom23940 - binom11920^22$$
The second solution uses a sum. It comes out to
$$sum_k=101^200 binomk+1920binom219-k20$$
combinatorics
combinatorics
edited Mar 16 at 13:30
YuiTo Cheng
2,1212837
2,1212837
asked Mar 16 at 8:46
David rossDavid ross
311
311
1
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Mar 16 at 8:51
$begingroup$
Hi, welcome to MSE! Could you explain us, please, what do you mean by "($X$ choose $Y$)"?
$endgroup$
– Yanior Weg
Mar 16 at 8:52
$begingroup$
@YaniorWeg $C^x_y$ that is I think he means combination.
$endgroup$
– Bijayan Ray
Mar 16 at 8:57
$begingroup$
Is not the number of possible ways to place 200 balls in 40 jars $binom23939$?
$endgroup$
– user
Mar 16 at 10:03
$begingroup$
@user Still waking up. Yes, you are correct.
$endgroup$
– N. F. Taussig
Mar 16 at 10:26
|
show 2 more comments
1
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Mar 16 at 8:51
$begingroup$
Hi, welcome to MSE! Could you explain us, please, what do you mean by "($X$ choose $Y$)"?
$endgroup$
– Yanior Weg
Mar 16 at 8:52
$begingroup$
@YaniorWeg $C^x_y$ that is I think he means combination.
$endgroup$
– Bijayan Ray
Mar 16 at 8:57
$begingroup$
Is not the number of possible ways to place 200 balls in 40 jars $binom23939$?
$endgroup$
– user
Mar 16 at 10:03
$begingroup$
@user Still waking up. Yes, you are correct.
$endgroup$
– N. F. Taussig
Mar 16 at 10:26
1
1
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Mar 16 at 8:51
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Mar 16 at 8:51
$begingroup$
Hi, welcome to MSE! Could you explain us, please, what do you mean by "($X$ choose $Y$)"?
$endgroup$
– Yanior Weg
Mar 16 at 8:52
$begingroup$
Hi, welcome to MSE! Could you explain us, please, what do you mean by "($X$ choose $Y$)"?
$endgroup$
– Yanior Weg
Mar 16 at 8:52
$begingroup$
@YaniorWeg $C^x_y$ that is I think he means combination.
$endgroup$
– Bijayan Ray
Mar 16 at 8:57
$begingroup$
@YaniorWeg $C^x_y$ that is I think he means combination.
$endgroup$
– Bijayan Ray
Mar 16 at 8:57
$begingroup$
Is not the number of possible ways to place 200 balls in 40 jars $binom23939$?
$endgroup$
– user
Mar 16 at 10:03
$begingroup$
Is not the number of possible ways to place 200 balls in 40 jars $binom23939$?
$endgroup$
– user
Mar 16 at 10:03
$begingroup$
@user Still waking up. Yes, you are correct.
$endgroup$
– N. F. Taussig
Mar 16 at 10:26
$begingroup$
@user Still waking up. Yes, you are correct.
$endgroup$
– N. F. Taussig
Mar 16 at 10:26
|
show 2 more comments
2 Answers
2
active
oldest
votes
$begingroup$
An expression for the number found by means of stars and bars is:
$$sum_k=0^99binomk+1919binom200-k+1919$$
We can rewrite this as:$$sum_i+j=238wedge ileq118binomi19binomj19$$under the convention that $binomnm=0$ if $mnotin0,1,dots,n$.
Further we have:$$binom23939=sum_i+j=238binomi19binomj19=$$$$sum_i+j=238wedge ileq118binomi19binomj19+sum_i+j=238wedge jleq118binomi19binomj19+binom11919^2$$where the first equality can be recognized as the hockey-stick equality.
This with:$$sum_i+j=238wedge ileq118binomi19binomj19=sum_i+j=238wedge jleq118binomi19binomj19$$
so that:$$sum_i+j=238wedge ileq118binomi19binomj19=frac12left[binom23939-binom11919^2right]$$
$endgroup$
$begingroup$
Nice. Both are equal to $514753813311660869268255952662500831886268154$. WolframAlpha can help (1 & 2).
$endgroup$
– Eric Duminil
Mar 16 at 11:23
$begingroup$
Thank you for the help
$endgroup$
– David ross
Mar 17 at 9:31
$begingroup$
You are very welcome.
$endgroup$
– drhab
Mar 17 at 11:48
add a comment |
$begingroup$
You made an error in applying "stars and bars". You should replace $binom23940$ with $binom23939$, and $binom11920$ with $binom11919$ and so on.
With correct expressions you obtain:
$$
sum_k=101^200 binomk+1919binom219-k19=fracbinom23939 - binom11919^22.
$$
No contradiction appears.
$endgroup$
$begingroup$
Thank you, I must have just gotten confused with the n and k notation
$endgroup$
– David ross
Mar 17 at 9:28
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3150203%2fhow-many-ways-can-200-identical-balls-be-distributed-into-40-distinct-jars%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
An expression for the number found by means of stars and bars is:
$$sum_k=0^99binomk+1919binom200-k+1919$$
We can rewrite this as:$$sum_i+j=238wedge ileq118binomi19binomj19$$under the convention that $binomnm=0$ if $mnotin0,1,dots,n$.
Further we have:$$binom23939=sum_i+j=238binomi19binomj19=$$$$sum_i+j=238wedge ileq118binomi19binomj19+sum_i+j=238wedge jleq118binomi19binomj19+binom11919^2$$where the first equality can be recognized as the hockey-stick equality.
This with:$$sum_i+j=238wedge ileq118binomi19binomj19=sum_i+j=238wedge jleq118binomi19binomj19$$
so that:$$sum_i+j=238wedge ileq118binomi19binomj19=frac12left[binom23939-binom11919^2right]$$
$endgroup$
$begingroup$
Nice. Both are equal to $514753813311660869268255952662500831886268154$. WolframAlpha can help (1 & 2).
$endgroup$
– Eric Duminil
Mar 16 at 11:23
$begingroup$
Thank you for the help
$endgroup$
– David ross
Mar 17 at 9:31
$begingroup$
You are very welcome.
$endgroup$
– drhab
Mar 17 at 11:48
add a comment |
$begingroup$
An expression for the number found by means of stars and bars is:
$$sum_k=0^99binomk+1919binom200-k+1919$$
We can rewrite this as:$$sum_i+j=238wedge ileq118binomi19binomj19$$under the convention that $binomnm=0$ if $mnotin0,1,dots,n$.
Further we have:$$binom23939=sum_i+j=238binomi19binomj19=$$$$sum_i+j=238wedge ileq118binomi19binomj19+sum_i+j=238wedge jleq118binomi19binomj19+binom11919^2$$where the first equality can be recognized as the hockey-stick equality.
This with:$$sum_i+j=238wedge ileq118binomi19binomj19=sum_i+j=238wedge jleq118binomi19binomj19$$
so that:$$sum_i+j=238wedge ileq118binomi19binomj19=frac12left[binom23939-binom11919^2right]$$
$endgroup$
$begingroup$
Nice. Both are equal to $514753813311660869268255952662500831886268154$. WolframAlpha can help (1 & 2).
$endgroup$
– Eric Duminil
Mar 16 at 11:23
$begingroup$
Thank you for the help
$endgroup$
– David ross
Mar 17 at 9:31
$begingroup$
You are very welcome.
$endgroup$
– drhab
Mar 17 at 11:48
add a comment |
$begingroup$
An expression for the number found by means of stars and bars is:
$$sum_k=0^99binomk+1919binom200-k+1919$$
We can rewrite this as:$$sum_i+j=238wedge ileq118binomi19binomj19$$under the convention that $binomnm=0$ if $mnotin0,1,dots,n$.
Further we have:$$binom23939=sum_i+j=238binomi19binomj19=$$$$sum_i+j=238wedge ileq118binomi19binomj19+sum_i+j=238wedge jleq118binomi19binomj19+binom11919^2$$where the first equality can be recognized as the hockey-stick equality.
This with:$$sum_i+j=238wedge ileq118binomi19binomj19=sum_i+j=238wedge jleq118binomi19binomj19$$
so that:$$sum_i+j=238wedge ileq118binomi19binomj19=frac12left[binom23939-binom11919^2right]$$
$endgroup$
An expression for the number found by means of stars and bars is:
$$sum_k=0^99binomk+1919binom200-k+1919$$
We can rewrite this as:$$sum_i+j=238wedge ileq118binomi19binomj19$$under the convention that $binomnm=0$ if $mnotin0,1,dots,n$.
Further we have:$$binom23939=sum_i+j=238binomi19binomj19=$$$$sum_i+j=238wedge ileq118binomi19binomj19+sum_i+j=238wedge jleq118binomi19binomj19+binom11919^2$$where the first equality can be recognized as the hockey-stick equality.
This with:$$sum_i+j=238wedge ileq118binomi19binomj19=sum_i+j=238wedge jleq118binomi19binomj19$$
so that:$$sum_i+j=238wedge ileq118binomi19binomj19=frac12left[binom23939-binom11919^2right]$$
answered Mar 16 at 10:21
drhabdrhab
103k545136
103k545136
$begingroup$
Nice. Both are equal to $514753813311660869268255952662500831886268154$. WolframAlpha can help (1 & 2).
$endgroup$
– Eric Duminil
Mar 16 at 11:23
$begingroup$
Thank you for the help
$endgroup$
– David ross
Mar 17 at 9:31
$begingroup$
You are very welcome.
$endgroup$
– drhab
Mar 17 at 11:48
add a comment |
$begingroup$
Nice. Both are equal to $514753813311660869268255952662500831886268154$. WolframAlpha can help (1 & 2).
$endgroup$
– Eric Duminil
Mar 16 at 11:23
$begingroup$
Thank you for the help
$endgroup$
– David ross
Mar 17 at 9:31
$begingroup$
You are very welcome.
$endgroup$
– drhab
Mar 17 at 11:48
$begingroup$
Nice. Both are equal to $514753813311660869268255952662500831886268154$. WolframAlpha can help (1 & 2).
$endgroup$
– Eric Duminil
Mar 16 at 11:23
$begingroup$
Nice. Both are equal to $514753813311660869268255952662500831886268154$. WolframAlpha can help (1 & 2).
$endgroup$
– Eric Duminil
Mar 16 at 11:23
$begingroup$
Thank you for the help
$endgroup$
– David ross
Mar 17 at 9:31
$begingroup$
Thank you for the help
$endgroup$
– David ross
Mar 17 at 9:31
$begingroup$
You are very welcome.
$endgroup$
– drhab
Mar 17 at 11:48
$begingroup$
You are very welcome.
$endgroup$
– drhab
Mar 17 at 11:48
add a comment |
$begingroup$
You made an error in applying "stars and bars". You should replace $binom23940$ with $binom23939$, and $binom11920$ with $binom11919$ and so on.
With correct expressions you obtain:
$$
sum_k=101^200 binomk+1919binom219-k19=fracbinom23939 - binom11919^22.
$$
No contradiction appears.
$endgroup$
$begingroup$
Thank you, I must have just gotten confused with the n and k notation
$endgroup$
– David ross
Mar 17 at 9:28
add a comment |
$begingroup$
You made an error in applying "stars and bars". You should replace $binom23940$ with $binom23939$, and $binom11920$ with $binom11919$ and so on.
With correct expressions you obtain:
$$
sum_k=101^200 binomk+1919binom219-k19=fracbinom23939 - binom11919^22.
$$
No contradiction appears.
$endgroup$
$begingroup$
Thank you, I must have just gotten confused with the n and k notation
$endgroup$
– David ross
Mar 17 at 9:28
add a comment |
$begingroup$
You made an error in applying "stars and bars". You should replace $binom23940$ with $binom23939$, and $binom11920$ with $binom11919$ and so on.
With correct expressions you obtain:
$$
sum_k=101^200 binomk+1919binom219-k19=fracbinom23939 - binom11919^22.
$$
No contradiction appears.
$endgroup$
You made an error in applying "stars and bars". You should replace $binom23940$ with $binom23939$, and $binom11920$ with $binom11919$ and so on.
With correct expressions you obtain:
$$
sum_k=101^200 binomk+1919binom219-k19=fracbinom23939 - binom11919^22.
$$
No contradiction appears.
answered Mar 16 at 10:26
useruser
5,82311031
5,82311031
$begingroup$
Thank you, I must have just gotten confused with the n and k notation
$endgroup$
– David ross
Mar 17 at 9:28
add a comment |
$begingroup$
Thank you, I must have just gotten confused with the n and k notation
$endgroup$
– David ross
Mar 17 at 9:28
$begingroup$
Thank you, I must have just gotten confused with the n and k notation
$endgroup$
– David ross
Mar 17 at 9:28
$begingroup$
Thank you, I must have just gotten confused with the n and k notation
$endgroup$
– David ross
Mar 17 at 9:28
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3150203%2fhow-many-ways-can-200-identical-balls-be-distributed-into-40-distinct-jars%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Mar 16 at 8:51
$begingroup$
Hi, welcome to MSE! Could you explain us, please, what do you mean by "($X$ choose $Y$)"?
$endgroup$
– Yanior Weg
Mar 16 at 8:52
$begingroup$
@YaniorWeg $C^x_y$ that is I think he means combination.
$endgroup$
– Bijayan Ray
Mar 16 at 8:57
$begingroup$
Is not the number of possible ways to place 200 balls in 40 jars $binom23939$?
$endgroup$
– user
Mar 16 at 10:03
$begingroup$
@user Still waking up. Yes, you are correct.
$endgroup$
– N. F. Taussig
Mar 16 at 10:26