Eleven people, 7 men and 4 women, successfully completed their study at the School of Astronautics. A team of 6 people needs to be selected from them for the next intergalactic travel. In how many ways can 6 astronauts be selected, if at least 2 women and 3 men must be in the team�H
This question was answered by Reiny and i was wondering how she got her andwer :
For at least 2 women and 3 men, the only cases would be
2W,4M = C(4,2) x C(7,4) = 6x35 = 210 , or
3W,3M = C(4,3) x C(7,3) = 4x35 = 140
for a total of 350 ways
I would like to know how you got the 35 in the 6 X 35 and the 4 X 35
Thank You
MathsHelpNeeded
Wow, this goes back quite some time.
by definition
C(n,r) = n!/(r!(n-r)!) , so
C(7,3) = 7!/(3!4!) = 35 , and
C(7,4) = 7!/(4!3!)
Most scientific calculators contain the functions C(n,r) and P(n,r)
they are probably written as nCr and nPr
Here is a page that explains the notation and difference between combination and permutation.
http://www.mathsisfun.com/combinatorics/combinations-permutations.html
You might also google "Pascal's Triangle" to see a very interesting relationship between the values of C(n,r) and the rows of the triangle.
Look at the last row of this version of the triangle
http://mathforum.org/workshops/usi/pascal/images/fill.comb.gif
To understand how Reiny arrived at the values of 35, let's break down the calculations:
1) Selecting 2 women and 4 men:
To calculate the number of ways to select 2 women from a group of 4 women, we use the combination formula C(n, r), which represents the number of ways to choose r items from a total of n items, without regard to the order. So, in this case, C(4, 2) gives us the number of ways to choose 2 women from 4 women.
Similarly, to calculate the number of ways to select 4 men from a group of 7 men, we use C(7, 4).
Multiplying these two values together (C(4, 2) x C(7, 4)) gives us the total number of ways to select 2 women and 4 men for the team.
2) Selecting 3 women and 3 men:
Using the same logic, C(4, 3) represents the number of ways to choose 3 women from 4 women, and C(7, 3) represents the number of ways to choose 3 men from 7 men.
Again, multiplying these values together (C(4, 3) x C(7, 3)) gives us the total number of ways to select 3 women and 3 men for the team.
The values 35, which appear in both cases, represent the results of the combination calculations: C(4, 2) = 6 and C(4, 3) = 4 (both equal to 6), and C(7, 4) = 35 and C(7, 3) = 35 (both equal to 35).
Therefore, when multiplying the number of ways to select women and men together, 6 x 35 gives the result of selecting 2 women and 4 men, and 4 x 35 gives the result of selecting 3 women and 3 men.