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# In how many ways can the crew of a ten oared boat be arranged,when of

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Math Expert
Joined: 02 Sep 2009
Posts: 57191
In how many ways can the crew of a ten oared boat be arranged,when of  [#permalink]

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10 Jul 2019, 01:10
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Difficulty:

95% (hard)

Question Stats:

38% (02:28) correct 62% (01:50) wrong based on 60 sessions

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In how many ways can the crew of a ten oared boat be arranged, when of the ten persons available, 2 of whom can row only on the bow side and 3 of whom can row only on the stroke side?

A. $$\frac{10!}{2!*3!}$$

B. $$\frac{10!}{8!*7!}$$

C. $$\frac{5!}{3!*2!}$$

D. $$\frac{(5!)^3}{3!*2!}$$

E. $$\frac{5!}{8!*7!}$$

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Re: In how many ways can the crew of a ten oared boat be arranged,when of  [#permalink]

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10 Jul 2019, 03:52
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Of the ten persons, 5 persons row on the bow side and 5 persons on the stroke side.

On the bow side, we already have 2 persons who only want to row on that side. Similarly, on the stroke side, we already have 3 persons who only want to row on that side.

This means, we are only left with 5 persons who do not have a preference towards rowing on a particular side. Of these 5 persons, any 2 can be selected to row on the stroke side. This will automatically ensure that the other 3 persons will row on the bow side.

Number of ways in which ANY 2 persons can be selected from 5 persons = $$5_C_2$$ = $$\frac{5!}{(3! * 2!)}$$.

Now, we need to take care of the arrangement part, since the question specifically asks about the number of ways in which the crew can be arranged.
On the bow side and the stroke side, the total number of arrangements possible is 5! and 5! respectively.

So, total number of ways of arranging the crew of 10 persons = $$\frac{5!}{(3! * 2!)}$$ * 5! * 5! = $$\frac{(5!)^3}{(3!*2!)}$$.
The correct answer option is D.

Hope this helps!
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Re: In how many ways can the crew of a ten oared boat be arranged,when of  [#permalink]

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28 Jul 2019, 10:29
Bunuel wrote:
In how many ways can the crew of a ten oared boat be arranged, when of the ten persons available, 2 of whom can row only on the bow side and 3 of whom can row only on the stroke side?

A. $$\frac{10!}{2!*3!}$$

B. $$\frac{10!}{8!*7!}$$

C. $$\frac{5!}{3!*2!}$$

D. $$\frac{(5!)^3}{3!*2!}$$

E. $$\frac{5!}{8!*7!}$$

The two additional persons to row on the stroke side can be chosen in 5C2 = 5!/(3!*2!) ways. Notice that once this choice has been made, all the remaining crew must row on the bow side; therefore, there is only one way to choose the 5 people to row on the bow side.

The five people to row on the stroke side can be arranged in 5! ways. Similarly, the five people to row on the bow side can also be arranged in 5! ways. Therefore, the boat can be crewed in

5!/(3!*2!) * 5! * 5! = (5!)^3/(3!*2!)

ways.

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Re: In how many ways can the crew of a ten oared boat be arranged,when of  [#permalink]

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02 Aug 2019, 14:36
ScottTargetTestPrep wrote:
Bunuel wrote:
In how many ways can the crew of a ten oared boat be arranged, when of the ten persons available, 2 of whom can row only on the bow side and 3 of whom can row only on the stroke side?

A. $$\frac{10!}{2!*3!}$$

B. $$\frac{10!}{8!*7!}$$

C. $$\frac{5!}{3!*2!}$$

D. $$\frac{(5!)^3}{3!*2!}$$

E. $$\frac{5!}{8!*7!}$$

The two additional persons to row on the stroke side can be chosen in 5C2 = 5!/(3!*2!) ways. Notice that once this choice has been made, all the remaining crew must row on the bow side; therefore, there is only one way to choose the 5 people to row on the bow side.

The five people to row on the stroke side can be arranged in 5! ways. Similarly, the five people to row on the bow side can also be arranged in 5! ways. Therefore, the boat can be crewed in

5!/(3!*2!) * 5! * 5! = (5!)^3/(3!*2!)

ways.

ScottTargetTestPrep could you please explain where is my thinking flawed?

I select the two persons to row on the stroke side using K!/(N-K)! since I thought that different orders would have given me different combinations, so:

5!/3! * 3! * 5! since I also need to consider the possible combinations of the other three people on the stroke side and the 5 people on the bow side.
I actually calculated the value but it doesn't match
Re: In how many ways can the crew of a ten oared boat be arranged,when of   [#permalink] 02 Aug 2019, 14:36
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