The probability of forming a 12-person committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor is:

Step 1: Calculate Total Possible Committees

First, we need to find the total number of ways to form a 12-person committee from the available 16 people (4 engineers + 2 doctors + 10 professors). We use the combination formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ to calculate this.

Step 2: Calculate Committees with Fewer than 3 Engineers

Next, we calculate the number of committees that do NOT satisfy the condition of having at least 3 engineers. This means committees with 0, 1, or 2 engineers. For each case, we choose engineers from the 4 available and the remaining members from the other 12 people (2 doctors + 10 professors).

Step 3: Calculate Committees with No Doctors

Now, we calculate the number of committees that do NOT satisfy the condition of having at least 1 doctor. This means committees with 0 doctors. We choose all 12 members from the 14 non-doctors (4 engineers + 10 professors).

Step 4: Calculate Committees with Fewer than 3 Engineers AND No Doctors

We need to find the number of committees that have fewer than 3 engineers AND no doctors. This means committees with 0, 1, or 2 engineers, chosen from the 4 engineers, and the remaining members chosen from the 10 professors (since there are no doctors). Note that $\binom{10}{12}$ and $\binom{10}{11}$ are 0, as you cannot choose more items than available.

Step 5: Calculate Favorable Committees using Principle of Inclusion-Exclusion

We use the Principle of Inclusion-Exclusion to find the number of committees that satisfy both conditions (at least 3 engineers AND at least 1 doctor). This is calculated as: Total committees - (committees with < 3 engineers) - (committees with no doctors) + (committees with < 3 engineers AND no doctors).

Step 6: Calculate the Probability

Finally, the probability is the ratio of the number of favorable committees to the total number of possible committees.

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