[Math] In how many ways can the letters of the word BANANA be rearranged such that the new word does not begin with a *B*

combinatorics

In how many ways can the letters of the word BANANA be rearranged such that the new word does not begin with a B?

What I did is incorrect. I said there are $5$ choices for the first place and then $5!$ possibilities after that for a total of $5\cdot5!=600$. However, I think I need to divide by $2$ and $3$ because of the repetitions of N and A. So how many ways can I do this? What am I missing?

Best Answer

For an alternative solution: choose the place for B, then choose the subset of places that contain an A, and this determines the word.

For B you have 5 options. For the set of positions with an A on them you have ${5}\choose{3}$ options.

Your total is $5*{{5}\choose{3}}=50$ words.

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