Question

In a hockey series between team X and Y, they decide to play till a team win $m$ match. Then the number of ways in which team X wins

• A. $2^m$
• B. ${}^{2m}{P}_m$
• C. ${}^{2m}{C}_m$
• D. None of these

Answer 1

Answer: (D)

None of these

Let 'W' indicate that India wins and let 'L' indicate that India loses.

Then, it is clear that in all the possible cases in which X wins, last match has to be won by X (as teams keep playing until any one of them wins $m$ matches).

So, all we have to do is permute $m-1$ W's with $0$ to $m-1$ loss (The $m$ win has to be on last position and loss can not exceed $m-1$ ).

${}_0^{m-1}C+{}_1^{m}C+{}_2^{m+1}C+{}_3^{m+2}C+{}_4^{m+3}C....................+{}_{m-1}^{2(m-1)}C={}_m^{2m-1}C$