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:
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Step-by-Step Solution
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 (kn)=kn!!(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 (1210) and (1110) 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.