Permutations And Combinations

Question
CBSEENMA11014205

What is the number of ways of choosing 4 cards from a pack of 52 cards? In how many of these :

(i) four cards are of the same suit?

(ii) four cards belongs to different suit?

(iii) are face cards?

(iv) two are red cards and two are black cards?

(v) cards are of the same colour? 

Solution

Number of cards in the deck = 52
Number of cards to be selected = 4
rightwards double arrow    Number of selections = space straight C presuperscript 52 subscript 4 equals fraction numerator 52 factorial over denominator 48 factorial cross times 4 factorial end fraction equals fraction numerator 52 cross times 51 cross times 50 cross times 49 over denominator 24 end fraction
                                    = 270725
(i) The four cards are of the same suit.
Options are:
They are all club cards
rightwards double arrow                  Number of selections = straight C presuperscript 13 subscript 4
                                       Or
They are all spade cards.
rightwards double arrow                  Number of selections = straight C presuperscript 13 subscript 4
                                                Or
They are all diamond cards.
rightwards double arrow                  Number of selections = straight C presuperscript 13 subscript 4
                                               Or
They are all heart cards.
rightwards double arrow                 Number of selections = straight C presuperscript 13 subscript 4
Hence, the total number of selections = <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre>
                                       
                                                 = 4 cross times fraction numerator 13 factorial over denominator 9 factorial space 4 factorial end fraction space equals space 4 cross times fraction numerator 13 cross times 12 cross times 11 cross times 10 over denominator 4 factorial end fraction equals 2860.
(ii) They are one from each suit.
rightwards double arrow 1 card is club and 1 card is spade and 1 card is diamond and 1 card is heart.
rightwards double arrow Total number of combinations = straight C presuperscript 13 subscript 1 cross times straight C presuperscript 13 subscript 1 cross times straight C presuperscript 13 subscript 1 cross times straight C presuperscript 13 subscript 1
                                             = 13 x 13 x 13 x 13
                                             = 28561.
(iii) The four cards are face cards.
                 Number of face cards = 4 kings + 4 queens + 4 jacks = 12
               Number of other cards  =  52 - 12 = 40
4 out of 12 face cards are to be selected
 
                        Number of selections = <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre>
(iv) There are 26 black cards and 26 red cards.
Two are to be selected from black cards and two from the red cards.
rightwards double arrow            Number of selections = space space straight C presuperscript 26 subscript 2 space end subscript cross times space straight C presuperscript 26 subscript 2 space equals space fraction numerator 26 factorial over denominator 24 factorial space 2 factorial end fraction cross times fraction numerator 26 factorial over denominator 24 factorial space 2 factorial end fraction
                                            = space space fraction numerator 26 cross times 25 over denominator 2 cross times 1 end fraction cross times fraction numerator 26 cross times 25 over denominator 2 cross times 1 end fraction space equals space 325 cross times 325
                                           
                                            =  105625
Hence, the number of selections = 105625
(v) The four cards are of the same colour.
rightwards double arrow   All four are either red or all four are black.
rightwards double arrow   Number of selections = straight C presuperscript 26 subscript 4 plus straight C presuperscript 26 subscript 4 space equals space 2 cross times straight C presuperscript 26 subscript 4
                                 
                                  = space space 2 times fraction numerator 26 factorial over denominator 22 factorial space 4 factorial end fraction equals space 2 cross times fraction numerator 26 cross times 25 cross times 24 cross times 23 cross times 22 factorial over denominator 22 factorial space space 4 factorial end fraction
                                 
                                 = fraction numerator 2 cross times 26 cross times 25 cross times 24 cross times 23 over denominator 4 cross times 3 cross times 2 cross times 1 end fraction equals 2 cross times 26 cross times 25 cross times 23
                                 = 29900.


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