Match each expression in Column-I with its correct value in Column-II. Column-I: (a) C(24,2) + C(23,2) + C(22,2) + C(21,2) + C(20,2) + C(20,3) (b) Number of ways to reach from (0,0) to (4,4) on a grid, passing through (2,2), moving only right or up one step at a time (c) Number of 4-digit

Question

Match each expression in Column-I with its correct value in Column-II. Column-I: (a) C(24,2) + C(23,2) + C(22,2) + C(21,2) + C(20,2) + C(20,3) (b) Number of ways to reach from (0,0) to (4,4) on a grid, passing through (2,2), moving only right or up one step at a time (c) Number of 4-digit numbers that can be formed using the digits {1, 2, 3, 4, 3, 2} (d) The largest natural number k such that 500! is divisible by 14^k Column-II: (p) 102, (q) 2300, (r) 82, (s) 36.

Accepted Answer

(a)->q; (b)->s; (c)->p; (d)->r. (a) C(24,2)+C(23,2)+...+C(20,2)+C(20,3): Using the hockey stick identity, sum_(k=r)ⁿ C(k,r) = C(n+1,r+1). So C(20,2)+C(21,2)+...+C(24,2) = C(25,3) - C(20,3) + C(20,3)... actually sumₖ₌₂₀²⁴ C(k,2) + C(20,3). By hockey stick: sumₖ₌₂²⁴ C(k,2) = C(25,3)=2300. But we need sum from k=20 to 24 plus C(20,3). sumₖ₌₂₀²⁴ C(k,2) = C(25,3) - C(20,3) [by hockey stick]. So total = C(25,3) - C(20,3) + C(20,3) = C(25,3) = 2300. (a)->q. (b) C(4,2)² = 36. (b)->s. (c) Digits: 1,2,2,3,3,4. 4-digit numbers: case all different: C(4,4)*4!=... choose 4 digits from {1,2,3,4} (taking one each of 2 and 3): all different: 4!/1=24 ways but we must pick which 4 from {1,2,3,4}: only one set

Solution

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