AndyB wrote:Hi All,

Need help in solving the below problem.
Q)Find the number of ways in which a selection of 4 letters can be made from the letters of the word "DISTILLATION".
a)2 alike, the other two distinct.

Ans)There are 13 letters of 8 different sorts I,I,,I,S,S,T,T,L,L,A,O,N,D.

My book gives the solution as -
The alike pair can be selected in 4 wasys.(Agreed)
While the other 2 distinct letters can be selected form the 7 distinct letters in 7C2 ways.
Hence the req number of ways are 4 x 7C2.

But i feel the solution is missing the point when (I,I) 2 alike is selected; we can still select the other 2 distinct from
(I,S,T,,L,A,O,N,D) eitht letters.So I think the solution given is wrong.

Could anyone please correct me if my understanding of the peoblem is wrong or explain me the correct answer.

Regards,

AndyB.

AndyB wrote:Hi GmatGuru,

The OA is the same given by you.

But my doubt is when we select I,I as the identical pair.
One extra I is left and then we can select 2 distinct letters from 8 letters.

At this point I get confused.Could you please help me in clarifying this confusion when I is selected as the like pair.

Regards,
AndyB.

2 identical letters:
III, TT, LL, SS give us 4 choices.

2 distinct letters:
From III, TT, LL, SS: Since 1 of these 4 letters was used above, we have 3 choices left.
The remaining 4 letters (D, A, O, N) give us another 4 choices.
Total number of choices = 3+4 = 7.
Number of combinations of 2 that can be formed from 7 choices = 7C2 = 21.

To combine our choices for the identical pair with our choices for the distinct pair, we multiply the results above:
4*21 = 84.