# Probability/ Combination Question | Help!

• Apr 5th 2010, 01:43 AM
mrsenim
Probability/ Combination Question | Help!
Hello,

I am new to probability.

Please have a look at the following question.

We have n = 10 persons and we wish to divide them at random into 3 groups consisting of 5, 3, and 2 persons respectively. In how many ways is this possible?

I have tried to solve it in following way.

Total number of possible ways =

$\displaystyle 10C5 = \left( 10! \over 5!(10-5)! \right) = \left( 10 \times 9 \times 8 \times 7 \times 6 \over 5 \times 4 \times 3 \times 2 \right) = \left( 30240 \over 120 \right) = 252$

Next!

$\displaystyle 5C3 = \left( 5! \over 3!(5-3)! \right) = \left( 5 \times 4 \over 2 \right) = \left( 20 \over 2 \right) = 10$

Next!

$\displaystyle 2C2 = \left( 2! \over 2!(2-2)! \right) = 1$

$\displaystyle S = 10C5 \times 5C3 \times 2C2 = 252 \times 10 \times 1 = 2520$

I think the above solution is correct but I really don't know how it happens.

I mean to say that I am confused in

1 - Why are we multiplying 10C5 with 5C3 and 2C2 ? Which rule or formula implies this multiplication of combinations?

2 - When and in which situations do I need to multiply combinations?

Any help would be highly appreciated.
• Apr 5th 2010, 02:42 AM
Quote:

Originally Posted by mrsenim
Hello,

I am new to probability.

Please have a look at the following question.

We have n = 10 persons and we wish to divide them at random into 3 groups consisting of 5, 3, and 2 persons respectively. In how many ways is this possible?

I have tried to solve it in following way.

Total number of possible ways =

$\displaystyle 10C5 = \left( 10! \over 5!(10-5)! \right) = \left( 10 \times 9 \times 8 \times 7 \times 6 \over 5 \times 4 \times 3 \times 2 \right) = \left( 30240 \over 120 \right) = 252$

Next!

$\displaystyle 5C3 = \left( 5! \over 3!(5-3)! \right) = \left( 5 \times 4 \over 2 \right) = \left( 20 \over 2 \right) = 10$

Next!

$\displaystyle 2C2 = \left( 2! \over 2!(2-2)! \right) = 1$

$\displaystyle S = 10C5 \times 5C3 \times 2C2 = 252 \times 10 \times 1 = 2520$

I think the above solution is correct but I really don't know how it happens.

I mean to say that I am confused in

1 - Why are we multiplying 10C5 with 5C3 and 2C2 ? Which rule or formula implies this multiplication of combinations?

2 - When and in which situations do I need to multiply combinations?

Any help would be highly appreciated.

Hi mrsenim,

$\displaystyle 10c_5=\binom{10}{5}$

is the number of ways of choosing a group of 5 from 10 available.

Taking one of these groups of 5, there will be 5 remaining.

$\displaystyle 5c_3=\binom{5}{3}$

is the number of ways to choose 3 from the remaining 5
(the 2 left over will always be unique to this 3).

Hence any selection of 5 from 10 can be matched with any one of these $\displaystyle 5c_3$ groups of 3 and 2.

Since there are $\displaystyle 10c_5$ groups of 5 in total,
we count the number of group matches by multiplying the number
of groups of 5 by the number of groups of 3.
• Apr 5th 2010, 09:19 PM
mrsenim
Quote:

Hi mrsenim,

$\displaystyle 10c_5=\binom{10}{5}$

is the number of ways of choosing a group of 5 from 10 available.

Taking one of these groups of 5, there will be 5 remaining.

$\displaystyle 5c_3=\binom{5}{3}$

is the number of ways to choose 3 from the remaining 5
(the 2 left over will always be unique to this 3).

Hence any selection of 5 from 10 can be matched with any one of these $\displaystyle 5c_3$ groups of 3 and 2.

Since there are $\displaystyle 10c_5$ groups of 5 in total,
we count the number of group matches by multiplying the number
of groups of 5 by the number of groups of 3.

But I am still confused about multiplying the two combination that is
$\displaystyle 10c_5 \times 5c_3$ to calculate the total number of group matches.

Could you please provide some link or documentation which clarifies this concept in detail?

Thanks Again.

Yours.
• Apr 5th 2010, 09:25 PM
Failure
Quote:

Originally Posted by mrsenim
But I am still confused about multiplying the two combination that is
$\displaystyle 10c_5 \times 5c_3$ to calculate the total number of group matches.

Could you please provide some link or documentation which clarifies this concept in detail?

What about Rule of product - Wikipedia, the free encyclopedia
• Apr 6th 2010, 02:43 AM
Hi mrsenim,

but here's a way to view it.

It's easier to understand with small groups.

Let's say we pick groups of 3 and 2 from a total of 6.

There are $\displaystyle 6c_2=15$ ways to pick 2.

Label these groups A, B, C, D, E, F, G, H, I, J, K, L, M, N, O.

Now take group A.
Having chosen the 2, there are 4 remaining from which to choose another different group of 2.

The second pair can be chosen in $\displaystyle 4c_2=6$ ways.

Label these groups U, V, W, X, Y, Z.

Group A may be paired with any of the groups U, V, W, X, Y, Z.
That's 6 pairings containing group A.

If we take the group B, we can form another 6 groups from the 4 remaining
after the 2 people in group B have been chosen.

Same for the other groups C to O.
Hence, there are 6 pairings 15 times.

Therefore there are (15)(6) pairings.

This is $\displaystyle \binom{6}{2}\binom{4}{2}$
• Apr 6th 2010, 09:14 PM
mrsenim
Quote:

Hi mrsenim,

Same for the other groups C to O.
Hence, there are 6 pairings 15 times.

Therefore there are (15)(6) pairings.

This is $\displaystyle \binom{6}{2}\binom{4}{2}$

ohh! ok. Thank you Archie Meade, it's clear now.

Quote:

Originally Posted by Failure

$\displaystyle {10\choose 5,3,2}={10!\over 5! 3! 2!}$