Miss Morrison has a test bank of multiple choice questions. 15 questions are combinations, and 12 are premutations. Miss Morrison is writing a test with 12 multiple choice questions. (Please include steps)

Question

a) How many different tests can she write if she wants to choose 7 combination and 5 premutation multiple choice questions.
b) After she has choosen the 12 questions, in how many orders can she put them on the test?

Answer 1

a) To choose 7 combination questions out of 15, we use the combination formula:

C(n, r) = n! / (r!(n-r)!)

where n is the total number of questions and r is the number of questions chosen. In this case, n = 15 and r = 7.

C(15, 7) = 15! / (7!(15-7)!)
= 15! / (7!8!)
= (15 * 14 * 13 * 12 * 11 * 10 * 9) / (7 * 6 * 5 * 4 * 3 * 2 * 1)
= 15 * 13 * 11 * 10 * 9
= 200,700

So, there are 200,700 different tests she can write if she wants to choose 7 combination and 5 permutation multiple choice questions.

To choose 5 permutation questions out of 12, we use the permutation formula:

P(n, r) = n! / (n-r)!

where n is the total number of questions and r is the number of questions chosen. In this case, n = 12 and r = 5.

P(12, 5) = 12! / (12-5)!
= 12! / 7!
= (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1)
= 792

So, there are 792 different tests she can write if she wants to choose 7 combination and 5 permutation multiple choice questions.

b) After she has chosen the 12 questions, the number of orders she can put them on the test is given by the permutation of 12 questions:

P(12, 12) = 12! / (12-12)!
= 12! / 0!
= 12!

So, she can put the 12 questions on the test in 12! = 12*11*10*9*8*7*6*5*4*3*2*1 = 479,001,600 different orders.