An instructor gives an exam with 14 questions. Students are allowed to choose an

Question

An instructor gives an exam with 14 questions. Students are allowed to choose any 10 to answer. (a) How many different choices of 10 questions are there? (b) Suppose 6 questions require proof and 8 do not. How many groups of 10 questions contain i. 4 that require proof and 6 that do not? ii. at least one that requires proof? iii. at most three that require proof? (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 10 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 10 questions are there?

Explanation / Answer

The number of ways of selecting r number of items out of n is given by nCr = 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

ans"

a. 1001, b. (i) 420, (ii) all 1001, (iii) 175, c. 506, d. 561

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