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# In How many ways can 7 people park cars in slots allotted to them?

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Manager
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In How many ways can 7 people park cars in slots allotted to them?  [#permalink]

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02 Jul 2016, 03:22
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65% (hard)

Question Stats:

52% (01:30) correct 48% (01:45) wrong based on 209 sessions

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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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Re: In How many ways can 7 people park cars in slots allotted to them?  [#permalink]

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02 Jul 2016, 04:51
5
susheelh wrote:
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

$$C^5_7=21$$: choosing which of the 7 persons will be on the slots allocates to them, the remaining 2 will be parked on each others slots.

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Intern
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Re: In How many ways can 7 people park cars in slots allotted to them?  [#permalink]

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25 Nov 2018, 21:44
I did not understand how answer is 21.
I calculated as:-
1x1x1x1x1x2C1x1x2!x3!=24
isnt this right? and why?
Intern
Joined: 16 Oct 2018
Posts: 8
Re: In How many ways can 7 people park cars in slots allotted to them?  [#permalink]

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28 Nov 2018, 21:27
Why do we use the combination formula instead of the permutation formula?
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Re: In How many ways can 7 people park cars in slots allotted to them?  [#permalink]

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28 Nov 2018, 23:28
1
Zynxu wrote:
Why do we use the combination formula instead of the permutation formula?

Hi Zynxu,

In how many ways can they park their cars so that exactly five persons park their cars in the slots allotted to them?

Or in other words, we can say in how many ways two parks will be parked in the wrong slot?

For example. car no. 6 and car no. 7 are parked in the wrong slot. This we won't count as 2, rather we will count as 1 way only.
Hence you need to apply combination.

Another way we can think as follows:

Suppose car no. 1 in in the wrong place, so all possible combination for the car no. 1 will be:

(1,2), (1,3), (1,4), (1,5), (1,6), (1,7) --- Total 6 ways.

For car no. 2, the following possibilities are there:

(2,1), (2,3), (2,4), (2,5) (2,6), (2,7) --- Total 6 ways.

Similarly, for all other cars, we'll have 6 such possibilities.

Hence the total number of ways = 7*6 = 42 ways.

But, we are double counting each option. Hence, we need to divide the final answer by 2.

The required number of ways = 42/2 = 21 ways.

Hope this helps.

Thanks.
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Re: In How many ways can 7 people park cars in slots allotted to them?  [#permalink]

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29 Nov 2018, 02:44
1
Pick any 5 person from 7 person = 5C7 = 7*6/3= 21 ways
These people parked on their respective slots.

Remaining 2 people & 2 Slots. only one way to park (i.e on other's parking space) possible. =1 way
Total possible ways= 21*1=21

Ans.B
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Re: In How many ways can 7 people park cars in slots allotted to them?  [#permalink]

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29 Nov 2018, 13:57
I did not understand how answer is 21.
I calculated as:-
1x1x1x1x1x2C1x1x2!x3!=24
isnt this right? and why?

Can you provide more information on your logic/intent behind that calculation?
Intern
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Posts: 1
Re: In How many ways can 7 people park cars in slots allotted to them?  [#permalink]

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29 Nov 2018, 19:02
7!/(7-5)!5!
This will give 21

Posted from my mobile device
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Re: In How many ways can 7 people park cars in slots allotted to them?  [#permalink]

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30 Nov 2018, 10:38
Bunuel wrote:
susheelh wrote:
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

$$C^5_7=21$$: choosing which of the 7 persons will be on the slots allocates to them, the remaining 2 will be parked on each others slots.

Super. I did the 7C5 but then i multiplied it by 2 thinking that the remaining cars can be parked in two ways whereas there is only one arrangement for them, as the last one would mean that they are in their correct places, which does not fit the answer. (a very confusing statement!!!)
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Re: In How many ways can 7 people park cars in slots allotted to them?  [#permalink]

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30 Nov 2018, 18:58
MayankSingh wrote:
Pick any 5 person from 7 person = 5C7 = 7*6/3= 21 ways
These people parked on their respective slots.

Remaining 2 people & 2 Slots. only one way to park (i.e on other's parking space) possible. =1 way
Total possible ways= 21*1=21

Ans.B

shouldn't it be 7C5????? and 42/2.... not 42/3......

☺☺

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In How many ways can 7 people park cars in slots allotted to them?  [#permalink]

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05 Dec 2018, 11:03
What I simply did was added the pairs of remaining 2 people who will have to park in each others spots:
Such as: 12,13,14,15,16,17,
23,24,25,26,27,
34,35,36,27,
45,46,47,
56,57,
67.

Hence option B. 21.
Notice that 12 is the same as 21, 23 is the same as 32, and so on. Hence, those need to be ignored.
Hope this helps
In How many ways can 7 people park cars in slots allotted to them?   [#permalink] 05 Dec 2018, 11:03
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