In how many ways can the letters in the word payment be arranged if the letters are taken 5 at a time?
in how many ways can the letters in the word butterfly be arranged if the letters are taken 5 at a time

This is a permutation problem in which you are to arrange the objects and the order doesn't matter. You can easily calculate it using your scientific calculator. Just use the formula nPr, where n is the total number of objects taken r at a time.

PAYMENT = 7 letters => 7P5 = 2520
There are 2,520 ways

For butterfly, its quite tricky because there are two letters. Oder does matter here, since no matter which t comes first, that is still the same. So, this is what you do:

BUTTERFLY = 9 letters
9P5 ÷ 2! => since there are two same letters
15120 ÷ 2 = 7560
There are 7560 ways.