How many different arrangements can be made with the letters in the word number?

The word NUMBER has 6 letters. This is a permutation with n = 6 elements:
P(6)= 6! = 6 · 5 · 4 · 3 · 2 ·1 = 720
720 different arrangements. 

38+105= 143, 39+104=143, and you will find then adding the numbers on opposite sides, moving inwards yields the answer 138 every time. because of this, you’re going to multiply 138 times the number of pairs, which is 34. therefore, 34*138= 4692(i’m sorry if it’s the wrong numbers, but it’s the right concept)