Permutations and Combinations Part 2

To find the total number of ways in which the middle letter is ' M ', we need to choose 2 letters from the set {A,B,C,...,L} and 2 letters from the set {N,O,P,...,Z}. The chosen letters must be arranged in alphabetical order with ' M ' in the middle.
Step by Step Solution:
Step 1
There are 12 letters before ' M ' ( A to L ) and 13 letters after ' M ' ( N to Z ).
Step 2
We need to choose 2 letters from the 12 letters before ' M '. This can be done in
(

12

2

)
ways.
Step 3
We need to choose 2 letters from the 13 letters after ' M '. This can be done in
(

13

2

)
ways.
Step 4
The total number of ways is the product of these combinations:
(

12

2

)×(

13

2

)=66×78=5148