Question

If five cards are chosen at random from a standard deck of playing cards, how many different ways are there to draw the five cards if at least three cards are a jack, queen or a king?

Answer 1

There are 4 each of Jacks, Queens and Kings making 12 cards. This leaves 52 - 12 = 40 other cards.
If 3 of the 5 drawn cards are Jack, Queen or King the number of combinations is 12C3. Each of these combinations can be taken with each of the 40C2 combinations of the other cards, making a total of 12C3 * 40C2.
If 4 of the 5 drawn cards are Jack, Queen or King the number of combinations is 12C4. Each of these combinations can be taken with each of the 40C1 combinations of the other cards, making a total of 12C4 * 40C1.
If 5 of the 5 drawn cards are Jack, Queen or King the number of combinations is 12C5.
Therefore the number of ways to draw 5 cards with at least 3 a Jack, Queen or King is:
\[(12C3\times40C2)+(12C4\times40)+12C5=can you\ calculate?\]

Answer 2

distribute?

Answer 3

There are various ways to calculate. My preference is to calculate each product and then add the results.

Answer 4

SO 12 times 40

Answer 5

960?

Answer 6

\[12C3\times40C2=\frac{12!\times40!}{3!\times9!\times2!\times38!}=\frac{12\times11\times10\times40\times39}{3\times2\times2}=you\ can\ calculate\]

Answer 7

2,059,200/12

Answer 8

Correct! Now do the division and you have the first product.

Answer 9

which is 171,600

Answer 10

thats how many ways?

Answer 11

Correct again! Now for the next product:
\[12C4\times40=\frac{12\times11\times10\times9\times40}{4\times3\times2}=you\ can\ calculate\]

Answer 12

ohh so 171,600

Answer 13

isnt the final answer?

Answer 14

No it's not.

Answer 15

19,800?

Answer 16

Correct! Now we find the value of 12C5 and add this value to 171600 + 19800.
\[12C5=\frac{12\times11\times10\times9\times8}{5\times4\times3\times2}=you\ can\ calculate\]

Answer 17

95,040 over 120

Answer 18

792??

Answer 19

Correct! So, finally, we have to add:
171600 + 19800 + 792 = ? ways

Answer 20

192,192

Answer 21

Correct again. Good work!

Answer 22

thanks

Answer 23

You're welcome :)

Answer 24

do you do tutoring

Answer 25

I tutor only on OpenStudy.

Answer 26

ok ive been struggling

Answer 27

Well I'll try to help when I'm logged in :)

Answer 28

ok

Answer 29

the words work can be arranged in how many different ways

Answer 30

24?

Answer 31

same with the word KINNIKINNIK

Answer 32

The letters in the word 'work' taken all at a time can be arranged in 4! = 24 ways.
There are 11 letters in the word KINNIKINNIK, K is repeated 3 times, I is repeated 4 times and N is repeated 4 times. The number of different ways of arranging the word KINNIKINNIK is:
\[\frac{11!}{3! \times4! \times4!}=\frac{11\times10\times9\times8\times7\times6\times5}{3\times2\times4\times3\times2}=you\ can\ calculate.\]

Answer 33

1,663,200/144?

Answer 34

= 11,550

Answer 35

The reason for dividing by
\[3! \times4! \times4!\]
is to deal with the repeated letters which can be arranged a number of times without making a new arrangement of the total letters.
Your result 11,550 ways is correct!

Answer 36

thats the final answer?

Answer 37

Yes, that's it!

Answer 38

ok cool

Answer 39

does P (6,2) = 30?

Answer 40

You're welcome. Please post any more questions as new ones, the reason being that I must log out now. BTW 30 is correct.

Answer 41

ok thanks :)

Answer 42

appreciate it your like the best on here lol

Answer 43

np