SOLUTION: A committee of 6 U.S. senators is to be formed with 3 Democrats and 3 Republicans. In how many ways can this be done if there are 61 Democratic senators and 39 Republican senators?

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Question 1152786: A committee of 6 U.S. senators is to be formed with 3 Democrats and 3 Republicans. In how many ways can this be done if there are 61 Democratic senators and 39 Republican senators?
Found 2 solutions by jim_thompson5910, ikleyn:
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!

We will use the combination function

to count the number of ways to select members. We use the combination function (not the permutation function) because order does not matter when selecting a committee. Any member does not outrank another, and there are no special positions. All that matters is the group as a whole rather an any particular individual.

Your book may use notation that looks like instead of , but its the same thing.

The exclamation marks represent factorials. Writing something like means so we start at 6 and count our way down to 1 multiplying along the way.

-----------------------------------------------

Note how something like is fully contained inside of the expression for

This means,

In short,

This will come in handy later on. Specifically the lines where I write "use the factorial trick" (see below). This is to cancel out factorials that would otherwise be massively large numbers.

-----------------------------------------------

For the Democrats, we have n = 61 people to choose from and r = 3 slots to fill.
Use the combination function to figure out how many ways there are to do this.



Replace n with 61, replace r with 3

Use the factorial trick. We started at 61 and stop at 58 because of the 58! in the denominator.

We have these highlighted terms pair up

and cancel out. A much simpler expression is left over

Expand out to get

At this point, we have three numbers multiplied in the numerator. It is not a coincidence this lines up with r = 3. So another shortcut is to start with the number n = 61 and count down by 1 until you have r = 3 items to multiply out in the numerator. Then you divide over 3! = 3*2*1

Let's simplify







There are 35990 ways to pick the Democrats.

We'll use this number later, so let A = 35990

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Repeat for the Republicans

n = 39

r = 3







Use the factorial trick

Terms pair up

and cancel out

Since r = 3, we have 3 numbers multiplied in the numerator. We started at n = 39 and counted down by 1 each time.

Expand out to get





Let B = 9139 represent the number of ways to pick the Republicans.
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To summarize what we found so far,
A = 35990 is the number of ways to pick the Democrats
B = 9139 is the number of ways to pick the Republicans

So,
A*B = 35990*9139 = 328,912,610 is the number of ways to pick the entire committee.
Order does not matter.
This number is very close to 329 million.

Answer by ikleyn(52790)   (Show Source): You can put this solution on YOUR website!
.

Usually, such problems are solved in three lines: 

    There are   =  = 35990 ways to select 3 democrats from 61 democrats.


    There are   =  = 9139 ways to select 3 republicans from 39 republicans.


    These selections are independent; therefore, the final ANSWER is the product of these numbers  35990*9139 = 328912610.

----------------

The symbols    denote  combinations  of  "n"  items taken  "k"  at a time.

On Combinations,  see introductory lessons
    - Introduction to Combinations
    - PROOF of the formula on the number of Combinations
    - Problems on Combinations
    - OVERVIEW of lessons on Permutations and Combinations
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic  "Combinatorics: Combinations and permutations".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.


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