A letter is taken out at random from ASSISTANT and another is taken out from STATISTICS. The probability that they are the same letters is:

Question: A letter is taken out at random from ASSISTANT and another is taken out from STATISTICS. The probability that they are the same letters is:

Solution:
The letters of ASSISTANT are A A S S S T T I N
The letters of STATISTICS are A S S S T T T I I C
Common letters are A, S, T, I
The probability of choosing an A from ASSISTANT & an A from STATISTICS is:
(²C₁/⁹C₁) × (¹C₁/¹⁰C₁)
= (2/9) × (1/10)
= 1/45

The probability of choosing an I from ASSISTANT & an I from STATISTICS is:
(¹C₁/⁹C₁) × (²C₁/¹⁰C₁)
= (1/9) × (2/10)
= 1/45

The probability of choosing a S from ASSISTANT & a S from STATISTICS is:
(³C₁/⁹C₁) × (³C₁/¹⁰C₁)
= (3/9) × (3/10)
= 1/10

The probability of choosing a T from ASSISTANT & a T from STATISTICS is:
(²C₁/⁹C₁) × (³C₁/¹⁰C₁)
= (2/9) × (3/10)
= 1/15

∴ The probability that the chosen letters are the same letters is:
1/45 + 1/45 + 1/10 + 1/15
= (2 + 2 + 9 + 6)/90
= 19/90