Suppose that a department contains 8 men and 16 women. How many ways are there to form a committee with 6 members if it must have strictly more women than men?

The various forms the committee can take:

WWWWWW (one way)
WWWWWM (6 ways — the M can be in any one of six chairs)
WWWWMM) (15 ways)

It cannot be 3W and 3 M (this would not be "strictly" more women)

Then in each combo, count the ways that each chair can be filled:

In the first case, from 16 women, pick 6
C(16, 6) = 16! / (6! 10!) = 8008

In the second case:
From 16 women pick 5 and from 8 men pick one
C(16,5)*C(8,1) = [16! / (5! 11!) ] * 8

In the third case:
From 16 women pick 4 and from 8 men pick 2
C(16,4)*C(8,2) = [16! / (4! 12!) ] * [ 8! / (2! 6!) ]

Add all three results.

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If the order is important (if the fact that a person is chosen for the 2nd chair instead of the 3rd chair makes a difference), then you’d have to worry about the various ways that each combo can be made:

There are 6 chairs and there are C(6. w) ways to arrange the six people (where w is the number of women on the committee).

C(6,6)*C(16,6)*C(8,0) + C(6,5)*C(16,5)*C(8,1) + C(6,4)*C(16,4)*C(8,2)

I added the C(8,0) to shorten the equation (which you will see later if you continue with probabilities).

C(8,0) means "from 8 men, pick zero" and there is only one way to pick none.
C(8,0) = 1
In fact, C(k,0)=1 for any positive number k.
Therefore, it changes nothing to the result, but shows a pattern

Total number of ways to pick a committee of 6 people, where the number of women can only be greater than the number of men, is a "sum"

Sum (as w goes from 4 to 6) of { C(6,w)*C(16, w)*C(8,(6-w))}

The "Sum as w…" part would be written as a capital sigma (the greek letter that looks like this Σ) with "w=4" written below the bottom line of the letter and "6" written above the top line of the letter.

It will simply mean, make a term for each possible value of w (here the possible values are 4, 5 and 6) then "sum" all these terms (meaning, add them up).

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Although it looks like an E for us, the letter Σ is an "S", and it stands for "sum" in this kind of equation.