# Intuition behind combinations

Example problem:

Suppose you have to select 5 cards from a deck: what is the probability of getting 4 diamonds and 1 spade.

The solution should be :

$$\ \frac{\binom{13}{4} \binom{13}{1}}{{52}\choose{5}}\$$

I have some understanding of how the formula works, i.e.that of all the possible 5-card combinations that chance can present me I should consider only those 4-card combinations of diamonds coupled with those 1-card combinations of spade.

I have no problem with the denominator. However, I am not capable of visualizing or have a clear intuition of what happens at the numerator level.

Do you have any suggestions for websites, books chapters or other sources which can give me some insights on problems of this kind?

• Have a look here. Personally for completeness and understanding I would rather write $$\frac{\binom{13}4\binom{13}1\binom{26}0}{\binom{52}{5}}$$There are $\binom{13}4$ ways to select $4$ diamonds out of $13$ diamonds, there are $\binom{13}1$ ways to select $1$ spade out of $13$ spades, There are $\binom{26}0=1$ ways to select $0$ hearts/clubs out of $13$ hearts/clubs. Multiplication gives you the number of ways of getting $4$ diamonds and $1$ spade (hence $0$ hearts/clubs). Apr 23, 2019 at 11:43