how many combinations of 4 officers can be elected from 10 members?

Question

use pascal's triangle

Answer 1

The answer is (10 choose 4) = 10!/(6!4!) = 210

This is the 5th entry from the left, of the 11th row of Pascal's triangle. That row starts out

1 10 45 120 210

Answer 2

it's 10C4 = 10!/(6!)(4!)

10*9*8*7/4*3*2*1= 210

Now pascals triangle starts with 1 and each entry is the sum of the 2 numbers above it.
since we want 10C4 we have to go 10 lines
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 256 210 120 45 10 1

That line represents combinations of ten objects taken
10 at a time =1
9 at a time =10
8 at a time = 45
7 at a time =120
6 at a time =210
5 at a time =256
4 at a time =210 <- this one
3 at a time =120
2 at a time =45
1 at a time =10
none at a time =1