JAMIA 2022 — Mathematics PYQ

1. Understanding the Winning Conditions

Suppose India wins the series. For India to win on the $n^{th}$ match, they must win exactly 4 of the previous $(n-1)$ matches, and then win the $n^{th}$ match.

The number of matches $n$ can be 5, 6, 7, 8, or 9.

2. Calculating Ways for India to Win

We calculate the combinations for each possible series length:

• Case 1: Series ends in 5 matches

  India wins all 5.

  $$\left(\genfrac{}{}{0ex}{}{4}{4}\right)=1\text{ way}$$

• Case 2: Series ends in 6 matches

  India wins 4 out of the first 5, and wins the $6^{th}$.

  $$\left(\genfrac{}{}{0ex}{}{5}{4}\right)=5\text{ ways}$$

• Case 3: Series ends in 7 matches

  India wins 4 out of the first 6, and wins the $7^{th}$.

  $$\left(\genfrac{}{}{0ex}{}{6}{4}\right)=15\text{ ways}$$

• Case 4: Series ends in 8 matches

  India wins 4 out of the first 7, and wins the $8^{th}$.

  $$\left(\genfrac{}{}{0ex}{}{7}{4}\right)=35\text{ ways}$$

• Case 5: Series ends in 9 matches

  India wins 4 out of the first 8, and wins the $9^{th}$.

  $$\left(\genfrac{}{}{0ex}{}{8}{4}\right)=70\text{ ways}$$

3. Total Ways for One Team

Total ways for India to win:

4. Total Ways for the Series

Since the question asks for the number of ways the series can be won (either by India or Pakistan), and both teams have an equal chance/structure of winning:

Final Answer

The total number of ways the series can be won is 252.