SOLUTION: From a standard deck of 52 cards, how many six-hand cards are possible if there are at least three aces and three other cards?

Question 1191612: From a standard deck of 52 cards, how many six-hand cards are possible if there are at least three aces and three other cards?

Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

If there are exactly 3 aces in the hand, then there are x = 4 ways to pick these three cards.
This is equivalent to having x = 4 ways to not pick a particular ace.

We have n = 52-4 = 48 non-ace cards to pick from and r = 3 slots to fill.
Use the nCr combination formula.
n C r = (n!)/(r!(n-r)!)
48 C 3 = (48!)/(3!*(48-3)!)
48 C 3 = (48!)/(3!*45!)
48 C 3 = (48*47*46*45!)/(3!*45!)
48 C 3 = (48*47*46)/(3!)
48 C 3 = (48*47*46)/(3*2*1)
48 C 3 = (103776)/(6)
48 C 3 = 17296
We have y = 17296 ways to select the three other non-ace cards.

In total, there are x*y = 4*17296 = 69,184 ways to select 6 cards such that exactly 3 are aces.

Let P = 69184

-------------------------------------

Now consider the six card hand having exactly 4 aces. There's only one way to select all four aces.

There are 48 C 2 = 1128 ways to pick the other cards.

That gives 1*1128 = 1128 ways to have a 6-card hand with exactly 4 aces.

Let Q = 1128

-------------------------------------

Add the previous results:
P+Q = 69184+1128 = 70,312
This is the number of six-card hands with at least 3 aces.
In the event of having 4 aces, we can't have "3 other cards" as your teacher mentioned because that would mean having a 7-card hand.

Answer: 70,312

RELATED QUESTIONS
A poker hand consists of 5 cards dealt from an ordinary deck of 52 playing cards. How... (answered by stanbon,edjones)

Three cards are drawn in succession and without replacement from a standard deck of 52... (answered by stanbon)

Three cards are selected at random from a standard deck of 52 playing cards. Determine... (answered by solver91311)

Three cards are selected at random from a standard deck of 52 playing cards. Determine... (answered by Edwin McCravy)

If four cards are chosen at random from a standard deck of playing cards, how many... (answered by stanbon)

A standard deck of cards has 52 cards in it. There are 4 cards of each number from 2 -... (answered by stanbon)

three cards are drawn in succession and without replacement from a standard deck of 52.... (answered by Edwin McCravy)

If six cards are chosen without replacement from a standard deck, how many hands are... (answered by richwmiller)

Two cards are drawn from a deck of 52 cards. If both cards are spades, how many different (answered by Boreal)