Kathren needs to plant a group of four different plants in each of her

Question

Kathren needs to plant a group of four different plants in each of her three flowerbeds. If she has twelve different plants, how many different arrangements of plants could she have in her garden?

A. 12
 B. 64
 C. 81
 D. 20736
 E. 34650

CrackverbalGMAT · Answer

For the first flower bed, we choose 4 out of 12 in  ^{12}C_4 = \frac{12 \space * \space * 11 \space * \space 10 \space * \space 9}{4 \space * \space * 3 \space * \space 2 \space * \space 1} = 495  ways

For the second flower bed, we choose 4 out of 8 remaining plants in  ^8C_4 = \frac{8 \space * \space * 7 \space * \space 6 \space * \space 5}{4 \space * \space * 3 \space * \space 2 \space * \space 1} = 70  ways

For the third flower bed, we choose 4 out of 4 remaining plants in  ^4C_4 = 1  way

Therefore the different number of arrangements = 495 * 70 * 1 = 34650

Alternatively, you can use the method of arrangements.

The number of ways of distributing n things equally among x things so that each gets an amount say y, then the total number of arrangements =  \frac{n!}{(y!)^x}

Here n = 12, x = 3 (flowerbeds) and y = 4 (plants in each flowerbed)

Therefore the total arrangements =  \frac{12!}{(4!)^3} = 34650

Option E

Arun Kumar

IanStewart · Answer

This is not a properly worded question. The word "arrangement" has a reserved meaning in combinatorics. You only ever talk about "arrangements" when you are putting things in order. But the start of the question suggests we're dividing the plants into "groups", from which I'd guess we're not putting the plants in order. There's no way to guess what the question is asking for. If I saw this in a real math book, I'd expect the answer to be 12!. That's far bigger than any answer choice. So I imagine the order of the flowers doesn't matter within each bed, in which case we're choosing 4 of 12 for the first bed, 4 of 8 for the second, and 4 of 4 for the last. 12C4 is clearly divisible by 5 (it equals (12)(11)(10)(9)/4!, so the '5' won't cancel out) so by units digits, the only plausible answer is E.

Kathren needs to plant a group of four different plants in each of her

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Kathren needs to plant a group of four different plants in each of her

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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