Binomial Theorem
Binomial coefficient sums — minimum p+q+m+n
MJAT_TS8_P1
Grade 12
Question:
Match each binomial sum in List-I with the minimum value of $p+q+m+n$ (where the sum $=\binom{p}{q}\cdot\binom{n}{m}$):
P) $\displaystyle\sum_{r=0}^{10}\binom{25}{r}\binom{10}{r}\binom{r+10}{r}$; Q) $\displaystyle\sum_{r=0}^{10}\binom{15}{10-r}\binom{25}{r}\binom{25-r}{10-r}$; R) $\displaystyle\sum_{r=0}^{16}(-1)^r\binom{25}{r}\binom{25-r}{9}\binom{16}{r}$; S) $\displaystyle\sum_{r=0}^{18}(-1)^r\binom{28}{r}\binom{r}{10}\binom{18}{r}$
List-II: 1)58; 2)60; 3)85; 4)70; 5)110
A) P→5; Q→4; R→3; S→2
B) P→5; Q→4; R→1; S→2
C) P→5; Q→4; R→1; S→3
D) P→3; Q→4; R→1; S→2
Step-by-Step Solution
Key Concept: Use Vandermonde-type and other binomial identities to simplify each sum to a product of two binomial coefficients, then find minimum $p+q+m+n$.
P→(3), Q→(4), R→(1), S→(2). Answer: **D**.
Correct Answer: D