(Each question 4 marks)

Question 14 Marks

What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these cards are of the same colour.

Answer

Given,there are 26 cards of red colour and 26 cards of black colour.
Thus, we have to select 4 cards either from black cards or red cards.
$\therefore$ Required number of ways $= ^{26}C_4 + ^{26}C_4$
$ =2 \times^{26} C_{4} $
$ =2 \times \frac{26 !}{4 ! 22 !} $
= 29900

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Question 24 Marks

What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these two are red cards and two are black cards.

Answer

Given,there are 26 red cards and 26 black cards.
Thus, the required number of ways = $^{26} \mathrm{C}_{2} \times^{26} \mathrm{C}_{2}$
$=\left(\frac{26 !}{2 ! 24 !}\right)^{2}=(325)^{2}=105625$

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Question 34 Marks

What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these are face cards.

Answer

Given,there are 12 face cards and we have to select 4 cards out of these.
$\therefore$ Required number of ways $= ^{12}C_4$
$ =\frac{12 !}{4 ! 8 !} $ = 495

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Question 44 Marks

What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these four cards belong to four different suits.

Answer

There are 13 cards in each suit. Thus, there are ${ }^{13} \mathrm{C}_1$ ways of choosing 1 card from 13 cards of diamond, ${ }^{13} \mathrm{C}_1$ ways of choosing 1 card from 13 cards of hearts, ${ }^{13} \mathrm{C}_1$ ways of choosing 1 card from 13 cards of clubs, ${ }^{13} \mathrm{C}_1$ ways of choosing 1 card from 13 cards of spades. Thus, by multiplication principle, therefore the required number of ways
$ = ^{13} C_{1} \times^{13} C_{1} \times^{13} C_{1} \times^{13} C_{1} $
$ =13 \times 13 \times 13 \times 13 $
$= 13^4$

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