Repeated alphabet in n^r
$begingroup$
Q: How many ways to form a word of 3 letters with English alphabet (repetition allowed)?
A: 26^3
but shouldn't we account for the repeated alphabet? like aab, and aab is the same?
Why isn't the question solved in the same way like "How many ways to arrange the word POOP"
which the answer is 4!/(2!2!)
permutations
$endgroup$
add a comment |
$begingroup$
Q: How many ways to form a word of 3 letters with English alphabet (repetition allowed)?
A: 26^3
but shouldn't we account for the repeated alphabet? like aab, and aab is the same?
Why isn't the question solved in the same way like "How many ways to arrange the word POOP"
which the answer is 4!/(2!2!)
permutations
$endgroup$
add a comment |
$begingroup$
Q: How many ways to form a word of 3 letters with English alphabet (repetition allowed)?
A: 26^3
but shouldn't we account for the repeated alphabet? like aab, and aab is the same?
Why isn't the question solved in the same way like "How many ways to arrange the word POOP"
which the answer is 4!/(2!2!)
permutations
$endgroup$
Q: How many ways to form a word of 3 letters with English alphabet (repetition allowed)?
A: 26^3
but shouldn't we account for the repeated alphabet? like aab, and aab is the same?
Why isn't the question solved in the same way like "How many ways to arrange the word POOP"
which the answer is 4!/(2!2!)
permutations
permutations
edited Dec 22 '18 at 7:35
thikin hax
asked Dec 22 '18 at 7:28
thikin haxthikin hax
11
11
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
We already are accounting for the repeated alphabet. That $26^3$ comes from answering three questions: What is the first letter? What is the second letter? What is the third letter?
Those three questions are allowed to have the same answers, as in your example "aab". That's $26$ possibilities for each question, and each answer is fully independent of the others.
Arranging the letters of a fixed word "deed"? We can ask which letter goes in each place, but those choices are tied together by the need to use each of that original word's letters. The numerator $4cdot 3cdot 2cdot 1$ comes from our choices being reduced at every step to whatever we haven't used yet, and the denominator $2times 2$ comes from letters of the same type being indistinguishable; $d_1e_1e_2d_2$ and $d_2e_1e_2d_1$ are the same words.
$endgroup$
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%2f3049192%2frepeated-alphabet-in-nr%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
$begingroup$
We already are accounting for the repeated alphabet. That $26^3$ comes from answering three questions: What is the first letter? What is the second letter? What is the third letter?
Those three questions are allowed to have the same answers, as in your example "aab". That's $26$ possibilities for each question, and each answer is fully independent of the others.
Arranging the letters of a fixed word "deed"? We can ask which letter goes in each place, but those choices are tied together by the need to use each of that original word's letters. The numerator $4cdot 3cdot 2cdot 1$ comes from our choices being reduced at every step to whatever we haven't used yet, and the denominator $2times 2$ comes from letters of the same type being indistinguishable; $d_1e_1e_2d_2$ and $d_2e_1e_2d_1$ are the same words.
$endgroup$
add a comment |
$begingroup$
We already are accounting for the repeated alphabet. That $26^3$ comes from answering three questions: What is the first letter? What is the second letter? What is the third letter?
Those three questions are allowed to have the same answers, as in your example "aab". That's $26$ possibilities for each question, and each answer is fully independent of the others.
Arranging the letters of a fixed word "deed"? We can ask which letter goes in each place, but those choices are tied together by the need to use each of that original word's letters. The numerator $4cdot 3cdot 2cdot 1$ comes from our choices being reduced at every step to whatever we haven't used yet, and the denominator $2times 2$ comes from letters of the same type being indistinguishable; $d_1e_1e_2d_2$ and $d_2e_1e_2d_1$ are the same words.
$endgroup$
add a comment |
$begingroup$
We already are accounting for the repeated alphabet. That $26^3$ comes from answering three questions: What is the first letter? What is the second letter? What is the third letter?
Those three questions are allowed to have the same answers, as in your example "aab". That's $26$ possibilities for each question, and each answer is fully independent of the others.
Arranging the letters of a fixed word "deed"? We can ask which letter goes in each place, but those choices are tied together by the need to use each of that original word's letters. The numerator $4cdot 3cdot 2cdot 1$ comes from our choices being reduced at every step to whatever we haven't used yet, and the denominator $2times 2$ comes from letters of the same type being indistinguishable; $d_1e_1e_2d_2$ and $d_2e_1e_2d_1$ are the same words.
$endgroup$
We already are accounting for the repeated alphabet. That $26^3$ comes from answering three questions: What is the first letter? What is the second letter? What is the third letter?
Those three questions are allowed to have the same answers, as in your example "aab". That's $26$ possibilities for each question, and each answer is fully independent of the others.
Arranging the letters of a fixed word "deed"? We can ask which letter goes in each place, but those choices are tied together by the need to use each of that original word's letters. The numerator $4cdot 3cdot 2cdot 1$ comes from our choices being reduced at every step to whatever we haven't used yet, and the denominator $2times 2$ comes from letters of the same type being indistinguishable; $d_1e_1e_2d_2$ and $d_2e_1e_2d_1$ are the same words.
answered Dec 22 '18 at 7:39
jmerryjmerry
9,4981124
9,4981124
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%2f3049192%2frepeated-alphabet-in-nr%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