India and South Africa play one day international series until one team wins $4$ matches. No match ends in a draw. Let k be the number of ways the series can be won.Find sum of digits of k ?

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Solution

Taking $I$ for India and $S$ for South Africa. We can arrange $I$ and $S A$ to shown the wins for India and South Africa respectively.
$e . g .$ : $I S S S S$ means first match is won by India which is followed by $4$ wins by South Africa. This is one way in which series can be won.
Suppose South Africa wins the series, then last match is always won by South Africa.
$∴$ Total number of ways $= 35$
In the same number of ways India can win the series.
$∴$ Total no. of ways in which the series can be won $35 \times 2 = 70$ .

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India and South Africa play one day international series until one team wins 4 matches. No match ends in a draw. Let k be the number of ways the series can be won.Find sum of digits of k ?

India and South Africa play one day international series until one team wins $4$ matches. No match ends in a draw. Let k be the number of ways the series can be won.Find sum of digits of k ?