Let’s count. Compositions with constraints

Do your best in trying to solve the following problems. It is normal that this step will take some time, we recommend to perseverate and wait before looking at the complete solution of the exercises in the video or in the pdf below.

Exercise 1. How many sequences formed with the letters (a, e, i, o, u, x, x, x, x, x, x, x, x,) (8 (x)’s) are there if one requires that no two vowels appear consecutively?

Hint: Chosen a 5-sequence of vowels, write (y_1) for the number of (x)’s to the left of the first vowel, then (y_2) for the number of (x)’s between the first and second vowels, …, (y_5) for the number of (x)’s between the fourth and fifth vowel, and finally (y_6) for the number of (x)’s appearing after the last vowel. Then (y_1+cdots+y_6=8) with (y_2,…, y_5geq 1).

Exercise 2. How many words can be formed using three (A)’s and no more than seven (B)’s?