SOLUTION: This exercise refers to a standard deck of playing cards. Assume that 5 cards are randomly chosen from the deck. How many hands contain 4 jacks?

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Question 1201443: This exercise refers to a standard deck of playing cards. Assume that 5 cards are randomly chosen from the deck.
How many hands contain 4 jacks?
Found 2 solutions by math_helper, math_tutor2020:
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!


The four jacks occupy four of the five positions in the hand. That leaves the fifth card as the only non-jack card.  Since this fifth card can be 48 different cards from the deck, there are 48 hands that have 4 jacks.  



Note that the order of selection does not matter (for example, selecting J,2h,J,J,J results in the same hand as  J,J,J,J,2h).

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

The tutor @math_helper has provided a great solution.

I'll mention a slightly different pathway.

We can use the nCr combination formula
  • x = 4C4 = 1 way to select the four jacks in any order.
  • y = 48C1 = 48 ways to select the fifth non-jack card
x*y = 1*48 = 48 different five-card hands that consist of all four jacks.

A few references about the nCr formula
https://mathworld.wolfram.com/Combination.html
https://www.mathsisfun.com/combinatorics/combinations-permutations.html
https://www.calculatorsoup.com/calculators/discretemathematics/combinations.php

Here is another question about card hands
https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1201482.html