How many strings of 9 lowercase letters from the Englishalph

How many strings of 9 lowercase letters from the Englishalphabet (of 26 letters) contain

a) the letter f at least once?

b)  the letters a, and x, in that order, with all letters distinct? For example, caebmxc is avalid string, because all letters are distinct, and a, and x appear in order within the string.

Solution

a) There are 26 alphabets

Hence there are 26 lowercase alphabets. If repitition is allowed,

from 9 lowercase letters total no of letters that can be formed = 26⁹

No of strings of lowercase letters with no f = No of 9 strings from the remaining 25 letters

= 25⁹

The remaining letters will be with atleast one string.

Strings of 9 lowercase letters from the Englishalphabet (of 26 letters) contain

the letter f at least once = 26⁹-25⁹

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b) Here all letters are distinct.

Also a and x should be in that order.

The remaining 7 letters can be selected from 24 alphabets excluding a and x in

24C7 ways.

In each letter we can insert a in the I place, and x in any of the remaining 8 gaps (including extreme ends)

So no of letters with a in the first place = 8

If a is inserted in the second place, x should be inserted in the remaining 7 gaps (including extreme ends of 6 alphabets) and so on.

End will be ax appearing at the last.

Thus no of possibilities = 8+7+6+...1 = 36 ways.

Total no of letters = 36(24C7)=346104(36)

= 12459744