A committee of 5 members is to be formed from a pool of 3 trainees, 4 professors, and 6 research associates. In how many different ways can the committee be formed if it must include exactly 2 trainees and 3 research associates?

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To determine the number of ways to form this committee, we need to calculate how many ways we can select the required members from each category and then find the total number of possible combinations.

**Step 1: Calculate the number of ways to select 2 trainees from 3 trainees.**
Using the combination formula: 
$$^nC_r = \frac{n!}{r!(n-r)!}$$

For selecting 2 trainees from 3:
$$^3C_2 = \frac{3!}{2!(3-2)!} = \frac{6}{2} = 3 	ext{ ways}$$

**Step 2: Calculate the number of ways to select 3 research associates from 6 research associates.**
$$^6C_3 = \frac{6!}{3!(6-3)!} = \frac{6 	imes 5 	imes 4 	imes 3!}{3! 	imes 3 	imes 2 	imes 1} = \frac{120}{6} = 20 	ext{ ways}$$

**Step 3: Apply the multiplication principle to find the total number of possible committees.**
Total number of ways = $^3C_2 	imes ^6C_3 = 3 	imes 20 = 60$

Therefore, the committee can be formed in 60 different ways.

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