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 if no match ends in a draw is

• A. 126
• B. 252
• C. 225
• D. None of these

Answer 1

Answer: (B)

Only five cases:
(i) all first five games win $\rightarrow 1$
(ii) $4$ out of first five and last one win $\rightarrow{ }^{5} C_{4}$
(iii) $4$ out of first six and last one win $\rightarrow{ }^{6} C _{4}$
(iv) $4$ out of first seven and last one win $\rightarrow{ }^{7} C _{4}$
(v) $4$ out of first eight and last one win $\rightarrow{ }^{8} C _{4}$
Total ways of winning the series
$=2 \times\left[1+{ }^{5} C_{4}+{ }^{6} C_{4}+{ }^{7} C_{4}+{ }^{8} C_{4}\right] $
$=2 \times{ }^{9} C_{5}=\frac{2 \times 9 \times 8 \times 7 \times 6 \times 5}{5 \times 4 \times 3 \times 2 \times 1}$
$=36 \times 7=252$