# Favorable choices

**Question**

Number of words formed using all of a,e,e,g,r such that a is adjacent to an e:

- 60
- 48
- 42
- 24

**Solution**

Let !!A!! be the set of all words such that !!ea!! occurs together. So we need to arrage, !!ea!!,!!e!!,!!g!!,!!r!!

The number of elements in this set are !!=4! =24!!

Let !!B!! be the set of all words such that !!ae!! occurs together. So we need to arrage, !!ae!!,!!e!!,!!g!!,!!r!!

The number of elements in this set are !!=4! =24!!

So we should just add the two? No, there are some cases which are counted in both. These are the cases where !!eae!! occurs.

So the number of such cases is !!=3! =6!!

So the number of favorable cases !!=24+24-6=42!!