A committee of 4 persons is to be formed from 2 ladies, 2 old men and 4 young men such that it includes at least 1 lady, at least 1 old man and at most 2 young men. Then the total number of ways in which this committee can be formed is :

We are given 2 ladies (L), 2 old men (O), and 4 young men (Y). We need to form a committee of 4 persons with the following conditions:

• At least 1 lady
• At least 1 old man
• At most 2 young men

Since the committee must have exactly 4 members and must include at least one lady and one old man, we consider all possible cases that satisfy the conditions. The young men can be 0, 1, or 2, but we must ensure the total is 4 and the minimums for ladies and old men are met.

The possible cases are:

1. 1 lady, 1 old man, 2 young men
2. 1 lady, 2 old men, 1 young man
3. 2 ladies, 1 old man, 1 young man
4. 2 ladies, 2 old men, 0 young men

We will calculate the number of ways for each case using combinations.

Case 1: 1 lady, 1 old man, 2 young men

Choose 1 lady out of 2: $$\binom{2}{1} = 2$$

Choose 1 old man out of 2: $$\binom{2}{1} = 2$$

Choose 2 young men out of 4: $$\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6$$

Total ways for this case: $$2 \times 2 \times 6 = 24$$

Case 2: 1 lady, 2 old men, 1 young man

Choose 1 lady out of 2: $$\binom{2}{1} = 2$$

Choose 2 old men out of 2: $$\binom{2}{2} = 1$$

Choose 1 young man out of 4: $$\binom{4}{1} = 4$$

Total ways for this case: $$2 \times 1 \times 4 = 8$$

Case 3: 2 ladies, 1 old man, 1 young man

Choose 2 ladies out of 2: $$\binom{2}{2} = 1$$

Choose 1 old man out of 2: $$\binom{2}{1} = 2$$

Choose 1 young man out of 4: $$\binom{4}{1} = 4$$

Total ways for this case: $$1 \times 2 \times 4 = 8$$

Case 4: 2 ladies, 2 old men, 0 young men

Choose 2 ladies out of 2: $$\binom{2}{2} = 1$$

Choose 2 old men out of 2: $$\binom{2}{2} = 1$$

Choose 0 young men out of 4: $$\binom{4}{0} = 1$$

Total ways for this case: $$1 \times 1 \times 1 = 1$$

Now, we add the number of ways from all cases to get the total number of committees:

Total ways = Case 1 + Case 2 + Case 3 + Case 4 = $$24 + 8 + 8 + 1 = 41$$

Hence, the total number of ways to form the committee is 41. Comparing with the options, A is 40, B is 41, C is 16, D is 32. Therefore, the correct answer is Option B.