Smart MATHEMATICAINS . I need Your help.?

Question

i know there are 4 aces, 4 kings and 4 queen but i can't sovle this question.
We pull 6 cards from a standard deck of 52.
(a) How many ways can we end up with all six cards being spades?

The answer for a is 1716 but i don't know how to get it.

(b) How many ways can we end up with two aces, three kings, and a queen?

The answer for b is 96 but i don't know how to get it

Answer 1

(a) there are 13 spades in a standard desk of 52 cards.

we can use combination nCr to calculate the number of ways.

number of ways with all 6 spades = 13C6 = 1716
remark:: nCr = n!/[(n-r)!r!]

(b) two aces, three kings and a queen
= 4C2*4C3*4C1
= 6*4*4
=96

Answer 2

a] there are a total of 13 spades in a deck of cards (52/4), since we want how many possible 6 card hands are made of spades, so we need to do a combination where n = 13 and r = 6. nCr = 13!/(6!(13-6)!) = 1716

b]this is similar to part a except do it for each one (two aces, 3 kings, 1 queen) then multiply like this:

Aces n = 4 r = 2 4C2 = 4!/(2!(4-2)!) = 6
Kings n = 4 r = 3 4C3 = 4!/(3!(4-3)!) = 4
Queen n = 4 r 1 4C1 = 4!/(1!(4-1)!) = 4

then multiply all three to get your 96

Answer 3

Hello,

a) there are 13 spades and 52 cards. So we have a combination of 13 things 6 at a time or 13C6= 13!/[(6!)*(13-6)!] or 13!/(6!*7!). If you are familiar with factorials this means that we have 13*12*11*10*.....1 on top or 6227020800 and on the bottom 6*5*4*...*1*7*6*5*....*1= 720*5040=362880 so we have 6227020800/362880 = 1716

b) We have two aces there are four aces so we have 4C2 and similarly three kings or 4C3 and a queen or 4C1 so we have

4!/[2!*2!] * 4!/[3!*1!]*4!/[1!*3!] = 6*4*4 = 96

Hope this helps!