Suppose that the U.S. Senate consisted of 57 Republican and 43 Democratic Senators. How many different 12-person committees could be formed with 8 Republican senators and 4 democratic senators?
C(57,8) x C(43,4) = ....
Is the answer 2508.52
and if so, how do I explane the whole solution. I am acting as a elementary teacher and have to discuss how to work each part out.
the answer has to be a whole number, you can't have part of a committee, (although with most governments that is obviously possible)
C(n,r) or nCr
is defined as n!/(r!(n-r)!)
and gives us the number of combinations of r things from n possibilites
e.g. given A, B, C, D, how many combination of 2 letters can we form
-we could list them AB, AC, AD, BC, BD, and CD
-notice this would be C(4,2) = 4!/(2!2!)
= 4x3x2x1/(2x1x2x1) = 24/4 = 6
a number like C(57,8) is quite large
around 1.6 billion
and to do it on your calculator by just using the definition, will overload your calculator.
Look for a key labeled something like
nCr
test it on C(4,2)
enter 4
press 2nd nCr
press 2
=
you should get 6
btw, the answer to your question is usually left in the notation I used, to express it in its full 15 digit number is rather futile.
Are you doing this in elementary school?? Wow
where?
To find the number of different 12-person committees that can be formed with 8 Republican senators and 4 Democratic senators, you can use the formula for combinations.
First, determine the number of ways you can choose the 8 Republican senators out of the 57 available Republican senators. This can be calculated using the combination formula:
C(n, r) = n! / (r! * (n-r)!)
Here, n represents the total number of items to choose from (57 Republican senators), and r represents the number of items you want to choose (8 Republican senators).
So, for the Republican senators, the calculation would be:
C(57, 8) = 57! / (8! * (57-8)!)
Next, determine the number of ways you can choose the 4 Democratic senators out of the 43 available Democratic senators using the same formula:
C(43, 4) = 43! / (4! * (43-4)!)
Finally, multiply the results of the two calculations together to find the total number of different 12-person committees that can be formed with 8 Republican senators and 4 Democratic senators:
Total number of committees = C(57, 8) * C(43, 4)
Simply plug in the values and calculate the expression to find the result.