The batting average for 20 innings of a cricket player is 25 runs. The highest score in an innings exceeds his lowest score by 86 runs. If these two innings are excluded the average score of the remaining 18 innings is 24 runs. Find his highest score in an innings.

The average score for all $$20$$ innings is $$25$$ runs.
Using $$\text{Average} = \dfrac{\text{Total runs}}{\text{Number of innings}}$$, the total runs in $$20$$ innings are

$$\text{Total}_{20} = 25 \times 20 = 500\ \text{runs}$$

Let the lowest score be $$L$$ and the highest score be $$H$$. We are told that

$$H - L = 86$$  $$-(1)$$

Removing these two innings leaves $$18$$ innings whose average is $$24$$ runs, so their total is

$$\text{Total}_{18} = 24 \times 18 = 432\ \text{runs}$$

The combined runs of the highest and lowest innings, therefore, equal the difference between the grand total and the remaining $$18$$ innings:

$$H + L = 500 - 432 = 68$$  $$-(2)$$

We now have the system of linear equations $$-(1)$$ and $$-(2)$$:

$$\begin{aligned} H - L &= 86 \\[4pt] H + L &= 68 \end{aligned}$$

Add the two equations to eliminate $$L$$:

$$2H = 86 + 68 = 154 \quad\Longrightarrow\quad H = \dfrac{154}{2} = 77$$

Hence, the highest score in an innings is $$77$$ runs.

Option C which is: 77