Question

A committe of four is being formed randomly of from the students and staff of a school: 5 administrators, 37 students and 4 teachers. How many ways can a committee be formed?

Answer 1

Since Committee consists of students and staff, it has to have at least one student and at least one Staff. Staff consists of 5 adminstrators and 4 teachers. The committee can be constituted as follows.

Student    Staff

1 1 Teacher 2 Adminstrators

   2 Teacher 1 Adminstrator

   3 Teachers

   3 Admindtrators

2 2 Teachers

2 Adminstratos

1 Teacher 1 Adminstrator

3 1 Teacher

1 Adminstrator

Number of ways in which committee can be formed:

1 S, 1T, 2 A = 37C₁x 4C₁ x5C₂ =37x4x(5x4)/(1x2) = 1480

1 S, 2T, 1A = 37C₁ x 4C₂ x 5C₁ = 37x(4x3)/(1x2) x5 =1110

1S, 3T =37C₁x 4C₃                =37 (4x3x2)/ (1x2x3) = 148

1S 3A = 37C₁ x5C₃    = 37x(5x4x3)/(1x2x3) = 370

2S 2T            =37C₂x 4C₂              = (37x36)/(1x2) x (4x3)/(1x2)=3996

2S 2A             = 37C₂x 5C₂            =(37x36)/(1x2) x(5x4)/(1x2)= 6660

2S 1T 1A      = 37C₂x 4C₁x5C₁ = (37x36)/(1x2) x4 x5 = 13320

3S 1T =37C₃x4C₁            = (37x36x35)/(1x2x3) x4=31080

3S 1A = 37C₃x 5C₁    = (37x36x35)/(1x2x3) x5 = 38850

    Total number of ways=1480+1110 +148+ 370 +3996+6660+13320+31080+38850=   97014                                                                                                                                                               

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