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)
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?
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.
a) To calculate the number of different tests Miss Morrison can write, we need to find the combinations and permutations separately.
For combinations:
Step 1: Find the number of ways to choose 7 out of 15 combination questions.
C(15, 7) = 15! / (7! * (15-7)!) = 6435
For permutations:
Step 1: Find the number of ways to choose 5 out of 12 permutation questions.
P(12, 5) = 12! / (12-5)! = 792
Step 2: Multiply the number of combinations and permutations together to get the total number of different tests.
Total = C(15, 7) * P(12, 5) = 6435 * 792 = 5,095,320 different tests.
b) After choosing the 12 questions, we need to find the number of possible orders for the test.
Step 1: Calculate the number of ways to arrange the 12 questions.
P(12, 12) = 12!
Step 2: Calculate the factorial of 12, which is 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 479,001,600.
Therefore, Miss Morrison can arrange the 12 questions in 479,001,600 different orders.