Simple applications of counting — practice questions

Question 1 · difficulty L2 · understanding

Let $p_n$ denote the total number of triangles formed by joining the vertices of an $n$-side regular polygon. If $p_{n+1} - p_n = 66$, then the sum of all distinct prime divisors of $n$ is :

• A. 7
• B. 8
• C. 5
• D. 6

Question 2 · difficulty L2 · understanding

The number of 3 -digit numbers, that are divisible by 2 and 3 , but not divisible by 4 and 9 , is _________.

Question 3 · difficulty L2 · understanding

The number of 5 digit numbers which are divisible by 4, with digits from the set {1, 2, 3, 4, 5} and the repetition of digits is allowed, is .................

Question 4 · difficulty L3 · understanding

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

• A. 56
• B. 16
• C. 24
• D. 48

Question 5 · difficulty L3 · understanding

Let P 1 , P 2 , ......, P 15 be 15 points on a circle. The number of distinct triangles formed by points P i , P j , P k such that i +j + k $$\ne$$ 15, is :

• A. 12
• B. 419
• C. 443
• D. 455

Question 6 · difficulty L3 · understanding

If the sides AB, BC and CA of a triangle ABC have 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to :

• A. 240
• B. 360
• C. 333
• D. 364

Question 7 · difficulty L3 · understanding

Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let $$\alpha$$ be the number of triangles having these points from different sides as vertices and $$\beta$$ be the number of quadrilaterals having these points from different sides as vertices. Then ($$\beta$$ $$-$$ $$\alpha$$) is equal to :

• A. 717
• B. 795
• C. 1890
• D. 1173

Question 8 · difficulty L3 · understanding

A natural number has prime factorization given by n = 2 x 3 y 5 z , where y and z are such that y + z = 5 and y $$-$$1 + z $$-$$1 = $${5 \over 6}$$, y > z. Then the number of odd divisions of n, including 1, is :

• A. 11
• B. 6
• C. 12
• D. 6x

Question 9 · difficulty L3 · understanding

Let m and $\mathrm{n},(\mathrm{m}<\mathrm{n})$, be two 2-digit numbers. Then the total numbers of pairs $(\mathrm{m}, \mathrm{n})$, such that $\operatorname{gcd}(m, n)=6$, is __________ .

Question 10 · difficulty L3 · understanding

If the number of seven-digit numbers, such that the sum of their digits is even, is $m \cdot n \cdot 10^n ; m, n \in\{1,2,3, \ldots, 9\}$, then $m+n$ is equal to__________