Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
In a particular batch of VIDYAMANDIR CLASSES Boston, there are 4 boys and certain number of girls. In every mock test only 5 students including at least 3 boys can appear. If different group of students write the Mock exam every time and number of times test conducted is 66 then find the total number of students in the class.
Step-by-Step Solution
Key Concept: The total number of distinct 5-student groups with at least 3 boys equals the sum of groups with exactly 3 boys and exactly 4 boys.
Let there be $g$ girls in the class. We need to form groups of 5 students with at least 3 boys from 4 boys total. The possible compositions are: (3 boys, 2 girls) or (4 boys, 1 girl). The number of ways to form such groups is $\binom{4}{3}\binom{g}{2} + \binom{4}{4}\binom{g}{1} = 4 \cdot \frac{g(g-1)}{2} + 1 \cdot g = 2g(g-1) + g = 2g^2 - g$. Setting this equal to 66: $2g^2 - g = 66$, which gives $2g^2 - g - 66 = 0$. Using the quadratic formula or factoring: $(2g + 11)(g - 6) = 0$, so $g = 6$. Therefore, total students = $4 + 6 = 10$.
Correct Answer: 10