The general formula for combinations is:
(n!)/(r!( n - r)!n!r!(n−r)!
Where nn is the number of objects and rr is how many are taken at a time.
n!n! means n x (n - 1) x ( n - 2 ).........( n - n + 2 )(n - n + 1):
This notation can seem confusing at first, but it basically just means the following:
5! => 5 xx 4 xx 3 xx 2 xx 15!⇒5×4×3×2×1
7! =>7 xx 6 xx 5 xx 4 xx 3 xx 2 xx 17!⇒7×6×5×4×3×2×1
etc.
Using formula:
n = 11n=11
r = 7r=7
(11!)/((7!(4)!11!(7!(4)!
We can make the calculation easier by cancelling first:
( 11 xx 10 xx 9 xx 8 xx 7 xx 6 xx 5 xx 4 xx 3 xx 2 xx 1)/(( 7 xx 6 xx 5 xx 4 xx 3 xx 2 xx 1 )( 4 xx 3 xx 2 xx 1))11×10×9×8×7×6×5×4×3×2×1(7×6×5×4×3×2×1)(4×3×2×1)
( 11 xx 10 xx 9 xx 8 xx cancel(7) xx cancel(6) xx cancel(5) xx cancel(4) xx cancel(3) xx cancel(2) xx cancel(1))/(( cancel(7) xx cancel(6) xx cancel(5) xx 4 xx 3 xx 2 xx 1 )( cancel(4) xx cancel(3) xx cancel(2) xx cancel(1)))
This leaves us:
(11 xx 10 xx 9 xx 8)/( 4 xx 3 xx 2 xx 1 ) = 7920/24 = color(blue)(330)
As a quick tip. Notice that what we are removing from the numerator is 7!
What remains in the denominator is #11 - 7 = 4.
This is the 4!
We know this at the start because we had the 11 and the 7'
These are the quick ways of evaluating by hand. A calculator is much quicker, but where's the fun in that :)
Hope this helps you.
