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 $\to 1$
(ii) $4$ out of first five and last one win $\to {}^5{C}_4$
(iii) $4$ out of first six and last one win $\to {}^6{C}_4$
(iv) $4$ out of first seven and last one win $\to {}^7{C}_4$
(v) $4$ out of first eight and last one win $\to {}^8{C}_4$
Total ways of winning the series
$=2\times \left[1+{}^5{C}_4+{}^6{C}_4+{}^7{C}_4+{}^8{C}_4\right]$
$=2\times {}^9{C}_5=\frac{2\times 9\times 8\times 7\times 6\times 5}{5\times 4\times 3\times 2\times 1}$
$=36\times 7=252$