Question

Find the number of ways in which 4 cards can be selected from a pack of 52 cards. In how many ways
(i) all 4 cards are from one suit?
(ii) all 4 cards are from different suits?
(iii) all 4 cards are face cards?
(iv) two cards are red and 2 cards are black?
(v) all 4 cards are of same colour?

Answer 1

No. of ways of selecting 4 cards from 52 cards `= .^(52)C_(4)`
`= (52 xx 51 xx 50 xx 49)/(1 xx 2xx 3 xx 4)`
`= 270725`
(i) No opf ways of selecting 4 cards of diamond from 14 cards `= .^(13)C_(4)`.
No of ways of selecting 4 cards of club from 13 cards `= .^(13)C_(4)`
No of ways of selecting 4 cards of hearts from 14 cards `= .^(13)C_(4)`
No of ways of selecting 4 cards of spade from 13 cards `= .^(3)C_(4)`
Now from addition principle
Required ways `=.^(13)C_(4) +^(13)C_(4) +^(13)C_(4) +^(13)C_(4)`
`= 4 xx .^(13)C_(4)`
`- 4xx (13 xx 12 xx 11 xx 10)/(1 xx 2 xx 3 xx 4)`
`= 2860`.
(ii) No. of ways of selecting one cardfrom 13 cards of diamond `= .^(13)C_(1) = 13`
No. of ways of selecting one card from 13 cards of clud `= .^(13)C_(1) = 13`
No of ways of selecting one card from 13 cards of spade `= .^(13)C_(1) = 13`
No of ways of selecting one card from 13 cards of heart `= .^(13)C_(1) = 13`
Now from multiplication principle
Required ways `= 13 xx 13 xx 13 xx 13`
`= 28561`
(iii) No of face cards in 52 cards = 12
No of ways of selecting 4 cards from 12 cards `= .^(12)C_(4)`
`= (12 xx 11 xx 10 xx 9)/(1 xx 2 xx 3 xx 4)`
`= 495`.
(iv) No. of ways of selecting 2 cards from 26 red cards `= .^(26)C_(2)`
No of ways of selecting 2 cards from 26 black cards `=.^(26)C_(2)` Now from multiplicarion principle
Required ways `= .^(26)C_(2) xx .^(26)C_(2)`
`= (26 xx 25)/(1 xx 2) xx (20 xx 25)/(1 xx 2)`
`= 105625`.
(v) No of ways of selecting 4 cards from 26 red `=.^(26)C_(4)`
Now from addition principle
Required ways ` .^(26)C_(4) +^(26)C_(4)`
`= 2 xx^(26)C_(4)`
`= 2xx (26 xx 25 xx 24 xx 23)`
`(1 xx 2xx 3 xx 4)`
`= 29900`.