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For example if a question is to find the number of different ways of arranging $4$ letters of $26$-letter alphabet with repetition, I know that we have to do $26^4$.

However, I am confused as to why exactly we are doing $26^4$. Are we assuming that all $4$ repetitions are of the same letter (For example 'AAAA' or 'BBBB')? Can't the letter repeat for $3$ times or $2$ times rather than $4$ times?

What if the word is something like 'AABC' or 'ABBC' etc?

Please explain in a way that an A-level student can understand. Thanks.

N. F. Taussig
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Hamza750
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2 Answers2

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"With repetition" means that repetition is allowed. Thus, in each of the four places, we have 26 choices of letter: $26^{4}$ possibilities.

We are not assuming any of the things you mention, and every possible combination of four letters is counted in the figure $26^{4}$. It means "any of the 26 letters can go in each of the four places."

Jonathan Hebert
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  • Do you mean to say that when repetition is allowed, our choices do not deplete like they do in permutations without repetition?? Like in the same question if we were to calculate the number of arrangements without repetition (26P4 = 262524*23)??? Am I right? – Hamza750 Apr 02 '15 at 21:20
  • Yes, that's what it means. – Jonathan Hebert Apr 02 '15 at 21:22
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Suppose _ _ _ _ these are four places and in each place you have to fill a letter (with repetition allowed). The first ' _ ' place can be filled by $26$ letters and next ' _ ' can also be filled by $26$ letters and so on till $4$th place. This will give you $26\times 26\times 26\times 26$ that is $26^4$.

Arnaud D.
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Aman Singh
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