Mixing
Evaluate $4^n = sum_{k=0}^{n} {n choose k} 3^k$
$begingroup$
Prove $4^n = sum_{k=0}^{n} {n choose k} 3^k$, using a combinatorial proof of
the set $S = {(a_1, a_2)| a_1, a_2 in {1...n}}$.
I'm having trouble figuring out how to prove $4^n$(LHS) using the set given.
combinatorics binomial-coefficients combinations combinatorial-proofs
edited Jan 18 at 16:08
N. F. Taussig
44.5k103357
$endgroup$
add a comment |
$begingroup$
Prove $4^n = sum_{k=0}^{n} {n choose k} 3^k$, using a combinatorial proof of
the set $S = {(a_1, a_2)| a_1, a_2 in {1...n}}$.
I'm having trouble figuring out how to prove $4^n$(LHS) using the set given.
combinatorics binomial-coefficients combinations combinatorial-proofs
edited Jan 18 at 16:08
N. F. Taussig
44.5k103357
$endgroup$
add a comment |
$begingroup$
Prove $4^n = sum_{k=0}^{n} {n choose k} 3^k$, using a combinatorial proof of
the set $S = {(a_1, a_2)| a_1, a_2 in {1...n}}$.
I'm having trouble figuring out how to prove $4^n$(LHS) using the set given.
combinatorics binomial-coefficients combinations combinatorial-proofs
edited Jan 18 at 16:08
N. F. Taussig
44.5k103357
$endgroup$
Prove $4^n = sum_{k=0}^{n} {n choose k} 3^k$, using a combinatorial proof of
the set $S = {(a_1, a_2)| a_1, a_2 in {1...n}}$.
I'm having trouble figuring out how to prove $4^n$(LHS) using the set given.
combinatorics binomial-coefficients combinations combinatorial-proofs
combinatorics binomial-coefficients combinations combinatorial-proofs
edited Jan 18 at 16:08
N. F. Taussig
44.5k103357
edited Jan 18 at 16:08
N. F. Taussig
44.5k103357
edited Jan 18 at 16:08
N. F. Taussig
44.5k103357
edited Jan 18 at 16:08
N. F. Taussig
44.5k103357
edited Jan 18 at 16:08
N. F. Taussig
44.5k103357
44.5k103357
asked Jan 18 at 15:24
user636032
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Proof: Consider all colourings of set {1,2,...,$n$} with colours red, green, blue, white. LHS is just a number of all of the colourings. To obtain RHS, first colour all numbers white, then choose $k$ of them, which will be recoloured with either red, green or blue each.
answered Jan 18 at 16:16
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
$endgroup$
$begingroup$
Please use the set given to prove the statement. For example S1 = {(1,1)}
$endgroup$
– user636032
Jan 19 at 23:37
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%2f3078380%2fevaluate-4n-sum-k-0n-n-choose-k-3k%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
$begingroup$
Proof: Consider all colourings of set {1,2,...,$n$} with colours red, green, blue, white. LHS is just a number of all of the colourings. To obtain RHS, first colour all numbers white, then choose $k$ of them, which will be recoloured with either red, green or blue each.
answered Jan 18 at 16:16
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
$endgroup$
$begingroup$
Please use the set given to prove the statement. For example S1 = {(1,1)}
$endgroup$
– user636032
Jan 19 at 23:37
add a comment |
$begingroup$
Proof: Consider all colourings of set {1,2,...,$n$} with colours red, green, blue, white. LHS is just a number of all of the colourings. To obtain RHS, first colour all numbers white, then choose $k$ of them, which will be recoloured with either red, green or blue each.
answered Jan 18 at 16:16
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
$endgroup$
$begingroup$
Please use the set given to prove the statement. For example S1 = {(1,1)}
$endgroup$
– user636032
Jan 19 at 23:37
add a comment |
$begingroup$
Proof: Consider all colourings of set {1,2,...,$n$} with colours red, green, blue, white. LHS is just a number of all of the colourings. To obtain RHS, first colour all numbers white, then choose $k$ of them, which will be recoloured with either red, green or blue each.
answered Jan 18 at 16:16
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
$endgroup$
Proof: Consider all colourings of set {1,2,...,$n$} with colours red, green, blue, white. LHS is just a number of all of the colourings. To obtain RHS, first colour all numbers white, then choose $k$ of them, which will be recoloured with either red, green or blue each.
answered Jan 18 at 16:16
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
answered Jan 18 at 16:16
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
answered Jan 18 at 16:16
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
answered Jan 18 at 16:16
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
2116
$begingroup$
Please use the set given to prove the statement. For example S1 = {(1,1)}
$endgroup$
– user636032
Jan 19 at 23:37
add a comment |
$begingroup$
Please use the set given to prove the statement. For example S1 = {(1,1)}
$endgroup$
– user636032
Jan 19 at 23:37
$begingroup$
Please use the set given to prove the statement. For example S1 = {(1,1)}
$endgroup$
– user636032
Jan 19 at 23:37
$begingroup$
Please use the set given to prove the statement. For example S1 = {(1,1)}
$endgroup$
– user636032
Jan 19 at 23:37
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%2f3078380%2fevaluate-4n-sum-k-0n-n-choose-k-3k%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
