Question

if you are dealt 4 cards from a shuffled deck of 52 cards, find the probability of getting one Queen and three king​

Answer 1

Answer: $\frac{16}{270725}$

This is the same as writing 16/270725

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Explanation:

There are 4 ways to draw a queen and 4 ways to select three kings (think of it like saying there are 4 ways to leave a king out). That produces 4*4 = 16 ways total to draw the four cards we want.

This is out of $_{52}C_4 = 270,725$ ways to select four cards without worrying if we got a queen and/or king. The steps to finding this number are shown below.

Divide the two values found to get the final answer $\frac{16}{270725}$

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Scratch Work:

Computing the value of $_{52}C_4$

Plug in n = 52 and r = 4 into the combination formula below

$_{n} C _{r} = \frac{n!}{r!*(n-r)!}\\\\_{52} C _{4} = \frac{52!}{4!*(52-4)!}\\\\_{52} C _{4} = \frac{52*51*50*49*48!}{4!*48!}\\\\_{52} C _{4} = \frac{52*51*50*49}{4!} \ \ \text{ .... the 48! terms cancel}\\\\_{52} C _{4} = \frac{52*51*50*49}{4*3*2*1}\\\\_{52} C _{4} = \frac{6,497,400}{24}\\\\_{52} C _{4} = 270,725$

We use the nCr combination formula (instead of the nPr permutation formula) because order doesn't matter with card hands.