a In how many ways can the letters of the word APPROXIMATION

a) In how many ways can the letters of the word APPROXIMATION be arranged? What would the answer be instead if the two Ps must be adjacent? (SHOW WORK & Explaination)

b) What is the coefficient of x¹⁰y²⁵ in the expansion of (2x+3y)³⁵? (SHOW WORK & Explaination)

Solution

The word \" APPROXIMATION\" is 13 letter word.

And in this word, we also have some repeating letters as well. That are given below:
A 2 times
P 2 times
O 2 times
I 2 times

So for removing the duplicates, we have to divide as well. Therefore the word can be arranged is:

13!
----------------- = 6227021000/16 = 38918883 Ways
2!*2!*2!*2!

If two P\'s are adjacent, then we have total 12 letters now. so the word can be arranged as

12!
------------ = 479001600/8 = 59875200 Ways
2!*2!*2!