Given the set S A B C D E how many ways are there to a sele

Given the set S = {A, B, C, D, E}, how many ways are there to a. select two letters from S if repetition is allowed? b. select two letters from S if repetition is not allowed? c. select three letters from S if repetition is allowed? d. select two letters from S such that the first letter is different from the second letter? e. select three letters from S such that no two letters in the three selected letters are the same?

Solution

Answers:

a) Total letters in S are 5 .

selecting 2 letters from S when repetitions allowed is 5² = 5x5 = 25

b) selecting 2 letters from S when repetition is not allowed is combination of 5 things taken 2 at a time i.e. ⁵C₂

₌ 5!/ 2!3! = 5x4/2 = 10

c) Total letters in S are 5 .

selecting 3 letters from S when repetitions allowed is 5³ = 5x5x5 = 125

d) Selecting 2 letters from S such that first letter is different from second :

First letter is A and second letter is any one of remainig four OR First letter is B and second letter is any one of remainig four OR First letter is C and second letter is any one of remainig four   OR First letter is D and second letter is any one of remainig four OR First letter is E and second letter is any one of remainig four

therefore ⁴C₁ + ⁴C_{1 +}⁴C_{1 +}⁴C_{1 +}⁴C_{1 =} 5.⁴C₁ = 5.4!/3! = 5.4 = 20

e) Select three letters from S such that no two letters are same:

First letter is A and second letter maybe any of four remaing letters and third may be any of remaning three letters.

In this way we have 5 possibilities possible .

Therefore total is = 5(⁴C₁.³C₁) = 5.4.3 = 60