Solve the following permutation question

Question

How many ways are there to select 11 countries in the United Nations to serve on a council if 1 is selected from a block of 51, 1 are selected from a block of 68 and 9 are selected from the remaining 70 countries?

Answer 1

This is not a permutation.  It's a combination.  The order the countries are picked in do not matter.  That is, if I'm selecting 9 from 70 countries, it won't matter if I pick country A first followed by country B, or country B first followed by country A.  They will still be A and B either way.  Therefore, order does not matter.  And for picking 1, it doesn't really matter anyway.

 

What does matter is separating the 3 parts, and doing each part individually.  The answers for each part will be multiplied together for a final answer.

 

There's 51 ways to pick 1 out of 51.  That will always be the case if you're only picking 1, but doing the math:

C(51,1) =     51!

                 1!(51-1)!

 

=     51!

    1!(50!)

 

=  1.55111875x10⁶⁶ 

    3.04140932x10⁶⁴

 

which does = 51

 

Likewise, there's 68 ways to pick 1 from 68, just like above.

 

Then you do C(70,9) = 70! / [9!(70-9)!]

 

(If you're having trouble working those out, just say so.)

 

Then the answers to each of those 3 are multiplied.  One way to know they're multiplied is the whole thing is "and"s.  That is, we pick the 1 and we pick the 1 and we pick the 9.  So it's like "continuing down the line" doing picking.  It's just that the first pick came from a pool of 51, and the next pick came from a pool of 68, and the next 9 picks all came from a pool of 70.

Answer 2

You have 11 selections to make:

 

First country, we have 51 options we will choose 1. Second country, 68 options, choose 1 and for the last 9 countries, we have 70 options, we will choose 9. Using the combination formula nCr = n!/[(n-r)!r!], we can calculate the following:

 

( 51 ) * ( 68 ) * ( 70 )   =   51!  *  68!  * 70!  

(  1  )    (  1  )    ( 9  )       50!1!  67!1!   61!9!

 

If my calculations are correct, the answer should be: 2.29 x 10¹⁴

Answer 3

For this question we are choosing r things from a pool of n, without repetition (ie, when a country is picked it cannot be picked again - the same country cannot fill more than one of the 11 council places), where the order does not matter. The number of ways to do this is n!/r!(n-r)!

 

First block: We need to select 1 country from 51. So r = 1 and n = 51. There are n!/r!(n-r)! ways to do this. In this case, this is 51!/1!50! = 51. (This is obvious from inspection - there are obviously 51 ways to choose one country from 51).

 

Second block. This is the same as the first, but r = 1 and n = 68. So there are 68!/67! = 68 ways.

 

Third block. Here r = 9 and n = 70. So there are 70!/9!(70-9)! ways, which is 70!/9!61!. This is 65,033,528,560.

 

Multiplying all these ways together, we get the total number of ways to pick 11 countries as specified from these 3 blocks = 51 x 68 x 70!/9!61!, which is approximately 2.26 x 10¹⁴.