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# How many different ways are there to arrange a group of 3 adults and 4

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Math Expert
Joined: 02 Sep 2009
Posts: 58335
How many different ways are there to arrange a group of 3 adults and 4  [#permalink]

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13 Dec 2016, 06:49
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Difficulty:

5% (low)

Question Stats:

89% (01:23) correct 11% (01:25) wrong based on 143 sessions

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How many different ways are there to arrange a group of 3 adults and 4 children in 7 seats if adults must have the first, third, and 7 seats?

A. 12
B. 144
C. 288
D. 1,400
E. 5,040

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Posts: 359
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Re: How many different ways are there to arrange a group of 3 adults and 4  [#permalink]

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13 Dec 2016, 07:56
There are 3 adults that should be seated on three particular seats and 4 children on the rest four. Adults and children cannot switch places with each other, only among themselves

Hence: $$3!*4! = 6*24 = 144$$

Intern
Joined: 12 Feb 2017
Posts: 4
How many different ways are there to arrange a group of 3 adults and 4  [#permalink]

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10 Dec 2017, 07:04
the adults have to be in 1 3 and 7th place , therefore : A*c*A*c*c*c*A
so it will be 3 x C x 2 x C x C x C x 1

now the children taking remaining places will be
3 x 4 x 2 x 3 x 2 x 1 x 1 = 144
Intern
Joined: 10 Dec 2017
Posts: 3
Re: How many different ways are there to arrange a group of 3 adults and 4  [#permalink]

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10 Dec 2017, 07:58
Bunuel wrote:
How many different ways are there to arrange a group of 3 adults and 4 children in 7 seats if adults must have the first, third, and 7 seats?

A. 12
B. 144
C. 288
D. 1,400
E. 5,040

Answer: 3 adults can be seated in 1st, 3rd and 7 th position in 3! ways. The remaining 4 positions are occupied by the remaining 4 people in 4! ways. Hence the total number of seating arrangements are 3!*4! = 144 ways.

Sneha Tatavarthy
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International Baccalaureate
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Re: How many different ways are there to arrange a group of 3 adults and 4  [#permalink]

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04 Jan 2018, 15:37
Hi All,

We're asked to arrange a group of 3 adults and 4 children in 7 seats with adults in the first, third, and seventh seats. We're asked for the number of arrangements possible. This question is a variation on a standard Permutation question. To solve it, we have to go from space to space and keep track of the 'options' available for each (noting that once we place a person, there is one fewer person available for the next equivalent spot).

For the 1st spot, there are 3 options
For the 2nd spot, there are 4 options
For the 3rd spot, there are 2 options
For the 4th spot, there are 3 options
For the 5th spot, there are 2 options
For the 6th spot, there are 1 options
For the 7th spot, there are 1 options

Total arrangements = (3)(4)(2)(3)(2)(1)(1) = 144 possible arrangments

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Re: How many different ways are there to arrange a group of 3 adults and 4   [#permalink] 04 Jan 2018, 15:37
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