Combinatorics Solved Examples

Example 1: Arrange the letters of the word TAMIL so that

• T is always next to L
• T and L are always together

Solution:

Let’s consider LT as one letter and keep it together. Now we have 4 alphabets that can be arranged in a row in 4⁢�4= 24. (General formula, nP_r)

L and T can be interchanged in their positions in 2! Ways. Therefore, total arrangements is given by 4!2! = 48.

Example 2: Calculate the number of ways a cricket eleven can be selected out of a batch of 15 players if;

• no restriction on the selection.
• A specific player is always selected.
• A specific player is never chosen.

Solution:

• When there is no restriction on the selection. This means 15c₁₁ gives you the total number of ways.
• Since a specific player is selected always. Therefore, 15 -1, we have 14c₁₀
• Since a specific player is never selected, we have 14c₁₁

Example 3: Calculate the number of committees of 5 students that can be chosen from a class of 25.

Solution:

Since we have to select 5 out of 25. Therefore,25⁢�5=25×24×23×22×215×4×3×2×1

= 53130