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Seven friends are living in 7 different flats of an apartmen

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Seven friends are living in 7 different flats of an apartmen

by alanforde800Maximus » Sun Aug 05, 2018 2:44 pm

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Seven friends are living in 7 different flats of an apartment. Each of them was allotted a parking slot to park their cars in the ground floor. But they used to park their cars randomly in any of the seven slots. In how many ways can they park their cars so that exactly five persons park their cars in the slots allotted to them?

A) 14
B) 21
c) 35
D) 42
E) 49
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by alanforde800Maximus » Sun Aug 05, 2018 2:49 pm
Can we solve above question via permutation or combination is the only way?
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by [email protected] » Sun Aug 05, 2018 6:17 pm
Hi alanforde800Maximus,

We're told that seven friends are living in 7 different flats of an apartment and each was allotted a parking slot to park their cars in the ground floor; however, they used to park their cars randomly in any of the seven slots. We're asked for the number of ways that they can park their cars so that EXACTLY five persons park their cars in the slots allotted to them. While complex-looking, this question is really just a Combination Formula question.

From a conceptual level, the only way for 5 friends to park in their assigned spots and 2 friends to NOT park in their assigned spots is when the group of 2 parks in each other's spots. For example, if everyone parked in their assigned spots, we would have:
ABCDEFG

If the first 5 friends parked in their assigned spots, but the other 2 didn't, we'd have:
ABCDEGF

Thus, you can either calculate the Combination of groups of 5 from a total of 7 OR groups of 2 from a total of 7. The result would be the same:
7!/(5!)(2!) or 7!/(2)(!5!) = (7)(6)/(2)(1) = 42/2 = 21

Final Answer: B

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