Discete math please help 1 show no matter how you list all 2

Discete math please help

1) show no matter how you list all 26 letters of alphabet there must be at least 14 consecutive constants.

Solution

In Eglish alphabets,

21 - Consonants

5 - vowels

Consider this problem as, there are total 26 slots available to arrange 26 alphabets. Condition is at least 14 alphabets must be consonants.

14 slots out of 26 can be selected in 26C14 ways. In these 14 selected slots any 14 consonants out of 21 can be arrranged in 21P14 ways.

In remaining 12 slots 12 aphabets can arranged in 12! ways.

Hence total number arrangements are = 26C14 * 21P14 * 12!