Permutations & Combinations
Permutations & Combinations
star_batch_jee_advanced_2025
Grade 11
Question:
An ice cream parlour has ice creams in eight different varieties. Number of ways of choosing $3$ ice creams taking atleast two are of the same variety, is:
(Assume that ice creams of the same variety to be identical & available in unlimited supply)
Step-by-Step Solution
Key Concept: Transform the sum using the identity $\frac{1}{r+1}\binom{n}{r} = \frac{1}{n+1}\binom{n+1}{r+1}$ to evaluate binomial sums.
Using the binomial coefficient identity, we express $\sum_{r=0}^{n} \frac{^nC_r}{r+1}b^r p^{r} = \frac{1}{n+1}\sum_{r=0}^{n} ^{n+1}C_{r+1}a^{n-r}b^r$ where $a = x^2$ and $b = 2 - x^2$. This simplifies to $\frac{1}{(n+1)(2-x^2)}[(a+b)^{n+1} - a^{n+1}]$. Substituting values and simplifying yields $\frac{2^{n+1} - x^{2n+2}}{(n+1)(2-x^2)}$.
Correct Answer: 2