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In how many ways can 10 people be divided into 4 groups so that 2 grou 30 Aug 2022, 01:30
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In how many ways can 10 people be divided into 4 groups so that 2 groups have 3 people each and 2 groups have 2 people each?

A. 5200
B. 6300
C. 8100
D. 3500
E. 10000


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In how many ways can 10 people be divided into 4 groups so that 2 grou 30 Aug 2022, 05:06
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Bunuel
In how many ways can 10 people be divided into 4 groups so that 2 groups have 3 people each and 2 groups have 2 people each?

A. 5200
B. 6300
C. 8100
D. 3500
E. 10000

Show more

So, we are creating 4 groups - two of 3 each and two of 2 each.

First group of 3 persons = 10C3 = 120
Next group of 3 persons out of remaining 7 = 7C3 = 35
Similarly next group of 2 persons out of remaining 4 = 4C2 = 6
Finally next 2 have to be selected out of remaining 2 = 2C2 = 1
Total ways = 120*35*6*1 = 25200 ways.

But the groups of 3 and groups of 2 are not distinguishable.
For example- If A, B, C and D are 4 groups in which A and B are of 3 each.
The way A(1,2,3), B(4,5,6), C(7,8) and D(9,10) will be same as A(4,5,6), B(1,2,3), C(7,8), D(9,10) but the calculation 25200 includes them as different.

To cater for repetitions, divide by 2!*2! => 25200/(2!*2!) = 6300
Two groups of 3 each are interchangeable in 2! ways and groups of 2 each are also interchangeable in 2! ways. Hence, 2!*2!
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In how many ways can 10 people be divided into 4 groups so that 2 grou 30 Aug 2022, 06:02
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chetan2u
Bunuel
In how many ways can 10 people be divided into 4 groups so that 2 groups have 3 people each and 2 groups have 2 people each?

A. 5200
B. 6300
C. 8100
D. 3500
E. 10000


So, we are creating 4 groups - two of 3 each and two of 2 each.

First group of 3 persons = 10C3 = 120
Next group of 3 persons out of remaining 7 = 7C3 = 35
Similarly next group of 2 persons out of remaining 4 = 4C2 = 6
Finally next 2 have to be selected out of remaining 2 = 2C2 = 1
Total ways = 120*35*6*1 = 25200 ways.

But the groups of 3 and groups of 2 are not distinguishable.
For example- If A, B, C and D are 4 groups in which A and B are of 3 each.
The way A(1,2,3), B(4,5,6), C(7,8) and D(9,10) will be same as A(4,5,6), B(1,2,3), C(7,8), D(9,10) but the calculation 25200 includes them as different.

To cater for repetitions, divide by 2!*2! => 25200/(2!*2!) = 6300
Show more

Hey, can you please explain why we are dividing twice with 2! to cater for repetitions? Thankyou

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In how many ways can 10 people be divided into 4 groups so that 2 grou 30 Aug 2022, 06:38
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chetan2u
Bunuel
In how many ways can 10 people be divided into 4 groups so that 2 groups have 3 people each and 2 groups have 2 people each?

A. 5200
B. 6300
C. 8100
D. 3500
E. 10000


So, we are creating 4 groups - two of 3 each and two of 2 each.

First group of 3 persons = 10C3 = 120
Next group of 3 persons out of remaining 7 = 7C3 = 35
Similarly next group of 2 persons out of remaining 4 = 4C2 = 6
Finally next 2 have to be selected out of remaining 2 = 2C2 = 1
Total ways = 120*35*6*1 = 25200 ways.

But the groups of 3 and groups of 2 are not distinguishable.
For example- If A, B, C and D are 4 groups in which A and B are of 3 each.
The way A(1,2,3), B(4,5,6), C(7,8) and D(9,10) will be same as A(4,5,6), B(1,2,3), C(7,8), D(9,10) but the calculation 25200 includes them as different.

To cater for repetitions, divide by 2!*2! => 25200/(2!*2!) = 6300

Hey, can you please explain why we are dividing twice with 2! to cater for repetitions? Thankyou

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Show more

Two groups of 3 each are interchangeable in 2! ways and groups of 2 each are also interchangeable in 2! ways. Hence, 2!*2!
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In how many ways can 10 people be divided into 4 groups so that 2 grou 23 Sep 2025, 08:21
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