Section 9.5 H 8. An instructor gives an exam with fourteen questions. Students a

Question

Section 9.5 H 8. An instructor gives an exam with fourteen questions. Students are allowed to choose any ten to answer. c. Suppose the exam instructions specify that at most one of questions 1 and 2 may be included among the ten. How many different choices of ten questions are there? d. Suppose the exam instructions specify that either both questions 1 and 2 are to be included among the ten or neither is to be included. How many different choices of ten questions are there? Section 9.6 H17. a. A store sells 8 kinds of balloons with at least 30 of each kind. How many different combinations of 30 balloons can be chosen? b. If the store has only 12 red balloons but at least 30 of each other kind of balloon, how many combinations of balloons can be chosen? c. If the store has only 8 blue balloons but at least 30 of each other kind of balloon, how many combinations of balloons can be chosen? d. If the store has only 12 red balloons and only 8 blue balloons but at least 30 of each other kind of balloon, how many combinations of balloons can be chosen?

Explanation / Answer

The number of ways of selecting r number of items out of n is given bynCr= n! / r!*(n-r)!

a) In this case n=14 and r=10,
nCr = 14C10 = 14!/10!*4! = 14*13*12*11*10! / 10!*4*3*2*1 = 1001

b) i) Number of groups containing 4 that require proof and 6 that do no = 6C4*8C6 = 420
ii) Since there are only 8 questions which do not require proof, all the possible groups (1001) will contain at least 1 that requires proof.
iii) Number of groups containing questions
2 requiring proof and 8 not requiring proof = 6C2*8C8 = 15
3 requiring proof and 7 not requiring proof = 6C3*8C7 = 160
Total = 15+160 = 175

NOTE: There will not be any groups containing only 1 question requiring proof because the number of questions not requiring proof is ONLY 8 (the group total becomes 9)

c and d answers are given BUT the questions are not typed.

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