Whoever wins 5 matches wins the series

B. 252

The minimum number of matches that the series can have is 5 matches. And this case is the easiest. The team winning the series, will win all 5 matches in a row i.e. W W W  W W, just 1 way.

If the series ends after 6 matches, the 6th match i.e. the last match will be won by the team winning the series … that is how the series ends. And amongst the first 5 matches, 4 matches will be won by the team winning the series and 1 match will be won by the team losing the series.

{W, W, W, W, L} W

Thus the outcome in the first 5 matches can be in ⁵C₁ ways.

If the series ends after 7 matches, the 7th match i.e. the last match will be won by the team winning the series … that is how the series ends. And amongst the first 6 matches, 4 matches will be won by the team winning the series and 2 matches will be won by the team losing the series.

{W, W, W, W, L, L} W

Thus the outcome in the first 6 matches can be in ⁶C₂ ways.

Proceeding the same way, we reach ….

If the series ends after 9 matches, the 9th match i.e. the last match will be won by the team winning the series … that is how the series ends. And amongst the first 8 matches, 4 matches will be won by the team winning the series and 4 matches will be won by the team losing the series.

{W, W, W, W, L, L, L, L} W

Thus the outcome in the first 8 matches can be in ⁸C₄ ways.

And one should realise that this is the longest that the series will play out i.e. by the 9th match, it is necessary that one of the teams wins 5 matches … it is not possible that each team would have won just a maximum of 4 matches.

And not to forget that the winning team could be either England or Australia.

Thus, answer = \(2 \times \left(1 \;+\; ^5C_1 \;+\; ^6C_2 \;+\; ^7C_3 \;+\; ^8C_4\right)\)

= 2 × (1 + 5 + 15 + 35 + 70) = 252