# In how many arrangements of the letters of the word PAPAYA are the three As consecutive?

Find the number of different arrangements that can be formed using all the letters of the word PAPAYA. Among these arrangements how many contain the three letters ‘A’s in consecutive order?

I have tried 6!/2!3!

• The number $\frac{6!}{2!3!}$ counts the number of distinguishable arrangements of the letters of the word PAPAYA. Hint: Think of AAA as a single letter. – N. F. Taussig Oct 23 '17 at 16:06

Youve got 3 A's. So there are $\binom{6}{3}$ ways you can put the A's in. After that, youve got 3 slots left, and two P's. So youve got $\binom{3}{2}$ ways to put the P's in. And then finally you have $\binom{1}{1}$ way to arrange the final Y. It doesnt matter the order you do these in, you could have started with the Y, and then the P and then the A's and instead of $\binom{6}{3}\binom{3}{2}\binom{1}{1}$ youd have $\binom{6}{1}\binom{5}{2}\binom{3}{3}$ and it ends up being the same answer, which is 60.
So of those 60 possible arrangements, have many have AAA? Well if we think of "AAA" as a single letter, now we have 4 slots to put the letters "AAA", "P", "P", and "Y" into, and no matter how we arrange them "AAA" will be together. So $\binom{4}{1} \binom{3}{2} \binom{1}{1}$