Question

In a unique hockey series between India and Pakistan, they decide to play on till a team wins $5$ matches. The number of ways in which the series can be won by India, if no match ends in a draw is :

• A. $126$
• B. $252$
• C. $225$
• D. $Noneofthese$

Answer 1

The correct option is A $126$

Let ${}^{\prime }{W}^{\prime }$ indicate that India wins and let ${}^{\prime }{L}^{\prime }$ indicate that India loses. Then, it is clear that in all the possible cases in which India wins, last match has to be won by India (as teams keep playing until any one of them wins 5 matches). So, all we have to do is permute 4 W's with 0,1,2,3 or 4 L's (The 5th W has to be in the last position and number of L's cannot exceed 4). i.e, all possible permutations of the left-hand side :

$WWWW|W$
$WWWWL|W$
$WWWWLL|W$
$WWWWLLL|W$
$WWWWLLLL|W$

So, answer

$=1+\frac{5!}{4!1!}+\frac{6!}{4!2!}+\frac{7!}{4!3!}+\frac{8!}{4!4!}$
$=1+5+15+35+70$
$=126$