Ten students, including Jack and Jil, are each entered into a race

Question

Ten students, including Jack and Jil, are each entered into a race to win one of five bicycles. If each student holds one entry in the race, what is the probability that Jack will win a bicycle but Jil will not?

A) 4/35
B) 4/15
C) 2/5
D) 49/105
E) 5/18

A) 4/35
 B) 4/15
 C) 2/5
 D) 49/105
 E) 5/18

KarishmaB · Answer

To win a bike, Jack should be in the top 5 and to not win a bike, Jill should be in the bottom 5.

For Jack, select a place in top 5 in 5 ways (1st/2nd/3rd/4th/5th)
For Jill, select a place in bottom 5 in 5 ways (6th/7th/8th/9th/10th)

For total cases, Jack has 10 spots available and Jill has 9 spots available.

Required Probability = 5*5/10*9 = 5/18

Answer (E)

tinytiger · Answer

P(Jack wins but Jill does not) = Total number of favorable outcomes / Total possible outcomes.

Total favorable outcomes = 5 x 5 (Jack picks any top 5 position while Jill picks any bottom 5) = 25.
Total number of possible outcomes = 10 x 9 (Jack picks 1 out of 10 possible slots, then Jill picks 1 out of the remaining 9) = 90.
P(Jack wins but Jill does not) = 25/90 = 5/18 (Option E).

Another way to do this is:
Total number of ways to select 5 out of 10 people to win 5 bicycles = 10C5
Total number of ways to select Jack but not Jill for 5 people = 8C4 (Jack and Jill are excluded from the pool; Jack must be selected, so we only need to pick 4 random persons out of the pool of 8 remaining candidates)
P(Jack wins but Jill does not) = 8C4 / 10C5 = 5/18 (Option E).

lfcelada · Answer

Why do you pick 3 and not 4 if it is a group of 5??

ccooley · Answer

I think this might just be a typo. 8c3 / 10c5 doesn't come out to 5/18, but 8c4 / 10c5 does. That also makes sense given what's in the problem: you need to choose FOUR other people, aside from Jack, who will win bikes.

tinytiger · Answer

Thanks for the spot, typo corrected for in my original post. Correct answer is to pick FOUR, not three (am avoiding the usage of numerals since I may have typed too quickly in my previous post).

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Ten students, including Jack and Jil, are each entered into a race

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Ten students, including Jack and Jil, are each entered into a race

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