Number of times I will get b balls from the bag
$begingroup$
I have 'n' identical balls and 'k' distinct bags. So total number of ways I can put balls with no restriction in bags is ${k+n-1 choose k-1}$. So I want to know the number of times there were exactly 'b' balls in a particular bag.
Eg-> I have 3 balls and 3 bags. So ${3+3-1 choose 3-1}$ = 10 ways.
If b = 0, there are 4 times [(0,0,3),(0,3,0),(0,1,2),(0,2,1)]
If b = 1, there are 3 times [(1,0,2),(1,2,0),(1,1,1)]
(here I am assuming that the chosen bag is first one)
So how many times do I get exactly 'b' balls in a particular bag?
combinatorics balls-in-bins
$endgroup$
add a comment |
$begingroup$
I have 'n' identical balls and 'k' distinct bags. So total number of ways I can put balls with no restriction in bags is ${k+n-1 choose k-1}$. So I want to know the number of times there were exactly 'b' balls in a particular bag.
Eg-> I have 3 balls and 3 bags. So ${3+3-1 choose 3-1}$ = 10 ways.
If b = 0, there are 4 times [(0,0,3),(0,3,0),(0,1,2),(0,2,1)]
If b = 1, there are 3 times [(1,0,2),(1,2,0),(1,1,1)]
(here I am assuming that the chosen bag is first one)
So how many times do I get exactly 'b' balls in a particular bag?
combinatorics balls-in-bins
$endgroup$
add a comment |
$begingroup$
I have 'n' identical balls and 'k' distinct bags. So total number of ways I can put balls with no restriction in bags is ${k+n-1 choose k-1}$. So I want to know the number of times there were exactly 'b' balls in a particular bag.
Eg-> I have 3 balls and 3 bags. So ${3+3-1 choose 3-1}$ = 10 ways.
If b = 0, there are 4 times [(0,0,3),(0,3,0),(0,1,2),(0,2,1)]
If b = 1, there are 3 times [(1,0,2),(1,2,0),(1,1,1)]
(here I am assuming that the chosen bag is first one)
So how many times do I get exactly 'b' balls in a particular bag?
combinatorics balls-in-bins
$endgroup$
I have 'n' identical balls and 'k' distinct bags. So total number of ways I can put balls with no restriction in bags is ${k+n-1 choose k-1}$. So I want to know the number of times there were exactly 'b' balls in a particular bag.
Eg-> I have 3 balls and 3 bags. So ${3+3-1 choose 3-1}$ = 10 ways.
If b = 0, there are 4 times [(0,0,3),(0,3,0),(0,1,2),(0,2,1)]
If b = 1, there are 3 times [(1,0,2),(1,2,0),(1,1,1)]
(here I am assuming that the chosen bag is first one)
So how many times do I get exactly 'b' balls in a particular bag?
combinatorics balls-in-bins
combinatorics balls-in-bins
edited Dec 13 '18 at 6:14
Sagar Chand
asked Dec 13 '18 at 4:38
Sagar ChandSagar Chand
1,2731520
1,2731520
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Your formula is slightly wrong. The number of ways in which $n$ balls can be put in $k$ distinct bags is $binom{n+k-1}{color{red}k-1}$. The reason being, that while applying stars and bars, we have to insert $k-1$ bars in any of $k+n-1$ positions, so that there are $k$ partitions formed, which we assume form the contents of each bag. For example, distributing $3$ balls in two boxes means you have to put $1$ bar between three balls, and there are $4$ places for it($2$ in the middle and two on either side) :
$$
- cdot-cdot-cdot-
$$
The problem that you state can be reformed as follows : if $x_i$ denotes the number of balls in bag $b_i$, then the number of ways that $n$ balls can be put in $k$ bags is the number of non-negative solutions to the equation $x_1+...+x_k = n$.
Suppose there are exactly $b$ balls in some pre-chosen bag, say $b_1$. Then, the rest of the balls contain exactly $n-b$ ball without restriction. So the number of possibilities is the number of solutions to $x_2+...+x_{k} = n-b$, the answer to which is $binom{n-b+k-2}{k-2}$, using the very same stars and bars logic with which one determines the number of non-negative solutions to $x_1+...+x_k = n$.
For example, if $n=3,k=3$, we see that $binom 52 = 10$.
If $b = 0$, then $binom{3-0+3-2}{3-2} = binom{4}{1} = 4$ possibilities present themselves. They are as you have specified.
If $b = 1$, then $binom{3-1+3-2}{3-2} = binom 31 = 3$ with possibilities as given.
$endgroup$
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%2f3037602%2fnumber-of-times-i-will-get-b-balls-from-the-bag%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Your formula is slightly wrong. The number of ways in which $n$ balls can be put in $k$ distinct bags is $binom{n+k-1}{color{red}k-1}$. The reason being, that while applying stars and bars, we have to insert $k-1$ bars in any of $k+n-1$ positions, so that there are $k$ partitions formed, which we assume form the contents of each bag. For example, distributing $3$ balls in two boxes means you have to put $1$ bar between three balls, and there are $4$ places for it($2$ in the middle and two on either side) :
$$
- cdot-cdot-cdot-
$$
The problem that you state can be reformed as follows : if $x_i$ denotes the number of balls in bag $b_i$, then the number of ways that $n$ balls can be put in $k$ bags is the number of non-negative solutions to the equation $x_1+...+x_k = n$.
Suppose there are exactly $b$ balls in some pre-chosen bag, say $b_1$. Then, the rest of the balls contain exactly $n-b$ ball without restriction. So the number of possibilities is the number of solutions to $x_2+...+x_{k} = n-b$, the answer to which is $binom{n-b+k-2}{k-2}$, using the very same stars and bars logic with which one determines the number of non-negative solutions to $x_1+...+x_k = n$.
For example, if $n=3,k=3$, we see that $binom 52 = 10$.
If $b = 0$, then $binom{3-0+3-2}{3-2} = binom{4}{1} = 4$ possibilities present themselves. They are as you have specified.
If $b = 1$, then $binom{3-1+3-2}{3-2} = binom 31 = 3$ with possibilities as given.
$endgroup$
add a comment |
$begingroup$
Your formula is slightly wrong. The number of ways in which $n$ balls can be put in $k$ distinct bags is $binom{n+k-1}{color{red}k-1}$. The reason being, that while applying stars and bars, we have to insert $k-1$ bars in any of $k+n-1$ positions, so that there are $k$ partitions formed, which we assume form the contents of each bag. For example, distributing $3$ balls in two boxes means you have to put $1$ bar between three balls, and there are $4$ places for it($2$ in the middle and two on either side) :
$$
- cdot-cdot-cdot-
$$
The problem that you state can be reformed as follows : if $x_i$ denotes the number of balls in bag $b_i$, then the number of ways that $n$ balls can be put in $k$ bags is the number of non-negative solutions to the equation $x_1+...+x_k = n$.
Suppose there are exactly $b$ balls in some pre-chosen bag, say $b_1$. Then, the rest of the balls contain exactly $n-b$ ball without restriction. So the number of possibilities is the number of solutions to $x_2+...+x_{k} = n-b$, the answer to which is $binom{n-b+k-2}{k-2}$, using the very same stars and bars logic with which one determines the number of non-negative solutions to $x_1+...+x_k = n$.
For example, if $n=3,k=3$, we see that $binom 52 = 10$.
If $b = 0$, then $binom{3-0+3-2}{3-2} = binom{4}{1} = 4$ possibilities present themselves. They are as you have specified.
If $b = 1$, then $binom{3-1+3-2}{3-2} = binom 31 = 3$ with possibilities as given.
$endgroup$
add a comment |
$begingroup$
Your formula is slightly wrong. The number of ways in which $n$ balls can be put in $k$ distinct bags is $binom{n+k-1}{color{red}k-1}$. The reason being, that while applying stars and bars, we have to insert $k-1$ bars in any of $k+n-1$ positions, so that there are $k$ partitions formed, which we assume form the contents of each bag. For example, distributing $3$ balls in two boxes means you have to put $1$ bar between three balls, and there are $4$ places for it($2$ in the middle and two on either side) :
$$
- cdot-cdot-cdot-
$$
The problem that you state can be reformed as follows : if $x_i$ denotes the number of balls in bag $b_i$, then the number of ways that $n$ balls can be put in $k$ bags is the number of non-negative solutions to the equation $x_1+...+x_k = n$.
Suppose there are exactly $b$ balls in some pre-chosen bag, say $b_1$. Then, the rest of the balls contain exactly $n-b$ ball without restriction. So the number of possibilities is the number of solutions to $x_2+...+x_{k} = n-b$, the answer to which is $binom{n-b+k-2}{k-2}$, using the very same stars and bars logic with which one determines the number of non-negative solutions to $x_1+...+x_k = n$.
For example, if $n=3,k=3$, we see that $binom 52 = 10$.
If $b = 0$, then $binom{3-0+3-2}{3-2} = binom{4}{1} = 4$ possibilities present themselves. They are as you have specified.
If $b = 1$, then $binom{3-1+3-2}{3-2} = binom 31 = 3$ with possibilities as given.
$endgroup$
Your formula is slightly wrong. The number of ways in which $n$ balls can be put in $k$ distinct bags is $binom{n+k-1}{color{red}k-1}$. The reason being, that while applying stars and bars, we have to insert $k-1$ bars in any of $k+n-1$ positions, so that there are $k$ partitions formed, which we assume form the contents of each bag. For example, distributing $3$ balls in two boxes means you have to put $1$ bar between three balls, and there are $4$ places for it($2$ in the middle and two on either side) :
$$
- cdot-cdot-cdot-
$$
The problem that you state can be reformed as follows : if $x_i$ denotes the number of balls in bag $b_i$, then the number of ways that $n$ balls can be put in $k$ bags is the number of non-negative solutions to the equation $x_1+...+x_k = n$.
Suppose there are exactly $b$ balls in some pre-chosen bag, say $b_1$. Then, the rest of the balls contain exactly $n-b$ ball without restriction. So the number of possibilities is the number of solutions to $x_2+...+x_{k} = n-b$, the answer to which is $binom{n-b+k-2}{k-2}$, using the very same stars and bars logic with which one determines the number of non-negative solutions to $x_1+...+x_k = n$.
For example, if $n=3,k=3$, we see that $binom 52 = 10$.
If $b = 0$, then $binom{3-0+3-2}{3-2} = binom{4}{1} = 4$ possibilities present themselves. They are as you have specified.
If $b = 1$, then $binom{3-1+3-2}{3-2} = binom 31 = 3$ with possibilities as given.
answered Dec 13 '18 at 5:50
астон вілла олоф мэллбэргастон вілла олоф мэллбэрг
38.5k33376
38.5k33376
add a comment |
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%2f3037602%2fnumber-of-times-i-will-get-b-balls-from-the-bag%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