A class consists of 15 boys of whom 5 are prefects. How many committees of 8 can be formed if each consists of at least 2 prefects?

There should be minimum $2$ perfects in the committee, there is no limit for maximum of perfects. So, The number of ways can be counted as:

$1$. $2$ perfect and $6$ non perfect..... Number of ways become ${{5}\choose{2}}\times{{10}\choose{6}}$.

$2$. $3$ perfect and $5$ non perfect....... Number of ways become ${{5}\choose{3}}\times{{10}\choose{5}}$.

$3$. $4$ perfect and $4$ non perfect.....Number of ways become ${{5}\choose{4}}\times{{10}\choose{4}}$.

$4$. $5$ perfect and $3$ non perfect.....Number of ways become ${{5}\choose{5}}\times{{10}\choose{3}}$.

Add all the ways and get the answer.

The solutions proposed by @JSchoone and @THELONEWOLF are correct.

You are counting committees with more than two prefects more than once.

There are $$\binom{5}{k}\binom{10}{8 - k}$$ ways to choose a committee with exactly $k$ prefects and $8 - k$ non-prefects. Adding the results for $2 \leq k \leq 5$ gives you the correct solution proposed by THE LONE WOLF. JSchoone's method of subtracting the number of committees with fewer than two prefects from the total is computationally more efficient.

In your method, when you choose two prefects and six more committee members from the remaining thirteen boys, your method counts each committee with three prefects $\binom{3}{2}$ ways, once for each of the three ways you could have chosen two of the three selected prefects as the two designated prefects. Your method counts each committee with four prefects $\binom{4}{2}$ ways, once for each of the six ways you could have chosen two of the four selected prefects as the two designated prefects. Your method counts the committee with all five prefects $\binom{5}{2}$ ways, once for each of the ten ways you could have designated two of the five selected prefects as the two designated prefects. Notice that $$\binom{5}{2}\binom{10}{6} + \binom{3}{2}\binom{5}{3}\binom{10}{5} + \binom{4}{2}\binom{5}{4}\binom{10}{4} + \binom{5}{2}\binom{5}{5}\binom{10}{3} = 17160$$

Also, notice that you are selecting the committee of eight boys from $5 + 13 = 18$ of the fifteen boys, while JSchoone and THE LONE WOLF are choosing the committee from the five prefects and non-prefects, that is, from the $10 + 5$ available boys. You ran into trouble because your group of five boys and group of thirteen boys are not separate groups.

The total number of committees of $8$ is $\binom{15}{8}$. The number of committees of $8$ with no prefects is equal to $\binom{10}{8}$. The number of committees with exactly one prefect is equal to $\binom{5}{1} \cdot \binom{10}{7}$.

Then the answer to your question is $\binom{15}{8} - \binom{10}{8} - \binom{5}{1} \cdot \binom{10}{7}=6435 - 45 - 5\cdot 120 = 5790.$