Match the counting results. Column I: (A) Maximum number of intersection points of five straight lines (B) Number of positive divisors of 2 4 ⋅ 3 5 ⋅ 5 3 (C) Number of triangles determined by five poi

Question

Match the counting results. Column I: (A) Maximum number of intersection points of five straight lines (B) Number of positive divisors of $2^{4} \cdot 3^{5} \cdot 5^{3}$ (C) Number of triangles determined by five points on a circle (D) Value of $\sum_{k = 1}^{n}_{n} P_{k} / k!$ Column II: (p) 120 (q) $2^{n} - 1$ (r) $(2 5) (s) (3 5)$

Column I	Column II
(A) Maximum number of intersection points of five straight lines	(p) 120
(B) Number of positive divisors of $2^{4} \cdot 3^{5} \cdot 5^{3}$	(q) $2^{n} - 1$
(C) Number of triangles determined by five points on a circle	(r) $(2 5)$
(D) Value of $\sum_{k = 1}^{n}_{n} P_{k} / k!$	(s) $(3 5)$

TestHub · Accepted Answer

Each pair of lines intersects once in the maximal arrangement, giving $(2 5) = 10$ . The divisor count is $5 \cdot 6 \cdot 4 = 120$ . Any three distinct points on a circle form a triangle, giving $(3 5) = 10$ . Finally $_{n} P_{k} / k! = (k n)$ , and the sum over $k = 1, \dots, n$ is $2^{n} - 1$ .

Note: the source left this key blank; mapping verified independently.

Mathematics - Permutation & Combination Question with Solution | TestHub

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