Permutation and Combination
Permutation and Combination
Allen Star Batch
Grade 11
Question:
An old man while dialing a 7 digit telephone number remembers that the first four digits consists of one 1's, one 2's and two 3's. He also remembers that the fifth digit is either a 4 or 5 while has no memory of the sixth digit, he remembers that the seventh digit is 9 minus the sixth digit. Maximum number of distinct trials he has to try to make sure that he dials the correct telephone number, is
360
240
216
None of these
Step-by-Step Solution
Key Concept: Symmetry of binomial coefficients combined with Vandermonde's identity relates sums of squared coefficients to central binomial coefficients.
Given the identity involving squared binomial coefficients $^{2n+1}C_0^2 + ^{2n+1}C_1^2 + ... + ^{2n+1}C_{2n}^2 = ^{2n+1}C_0^2 + ^{2n+1}C_1^2 + ... - ^{4n+2}C_{2n+1}$, we rearrange to get $\sum_{r=0}^{2n} {}^{2n+1}C_r^2 = \sum_{r=0}^{2n} {}^{2n+1}C_{2r+1} = \frac{1}{2}(^{4n+2}C_{2n+1})$.
Correct Answer: 2