Question

A committee of \(8\) members is to be formed from \(5\) teaching staff, \(4\) office staff and \(6\) students so as to include at least two from each category. Then the total number of ways of forming the committee is

• A. \(4100\)
• B. \(3950\)
• C. \(3200\)
• D. \(3500\) \bigskip

Hint:

For committee formation problems with minimum restrictions, first distribute the required number of members among the categories, then count each case using combinations and add the results.

Answer 1

The Correct Option is A

Concept: Distribute the committee members among the three groups while satisfying the minimum requirement, then use combinations.

Step 1: The valid distributions are \((2,2,4),(2,3,3),(2,4,2),(3,2,3),(3,3,2)\) and \((4,2,2)\).

Step 2: Their respective counts are \(900,800,150,1200,600\) and \(450\).

Step 3: Adding all cases, \(900+800+150+1200+600+450=4100\). Hence, \(\boxed{4100}\).