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# Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5

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Math Expert
Joined: 02 Sep 2009
Posts: 54401
Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5  [#permalink]

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24 Mar 2019, 23:57
00:00

Difficulty:

55% (hard)

Question Stats:

50% (03:09) correct 50% (03:28) wrong based on 14 sessions

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Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let p be the probability that all four slips bear the same number. Let q be the probability that two of the slips bear a number a and the other two bear a number $$b \neq a$$. What is the value of q/p?

(A) 162
(B) 180
(C) 324
(D) 360
(E) 720

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Math Expert
Joined: 02 Aug 2009
Posts: 7575
Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5  [#permalink]

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25 Mar 2019, 06:14
Bunuel wrote:
Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let p be the probability that all four slips bear the same number. Let q be the probability that two of the slips bear a number a and the other two bear a number $$b \neq a$$. What is the value of q/p?

(A) 162
(B) 180
(C) 324
(D) 360
(E) 720

Since, we have probability out of the same set, and we have probability in both numerator and denominator, the q/p can be taken as (ways of picking two different pairs of number)/(ways of picking all four same). This will save some calculations..

Ways to choose all four - 10 *$$\frac{4!}{4!}$$, as all the numbers are similar, so 10 ways
Ways to choose two different pair - 10C2*4C2*4C2, that is (choose two out of 10), then there are 4 of each of these number, so 4C2 ways for each number)

Thus $$\frac{q}{p}=\frac{10C2*4C2*4C2}{10}=\frac{10*9*4*3*4*3}{2*2*2*10}=9*9*2=162$$
A
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Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5   [#permalink] 25 Mar 2019, 06:14
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# Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5

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