Question

Arrange the following values in descending order:
a. The number of diagonals of a decagon
b. The number of positive integral factors of 2940
c. The number of circles that can be drawn out of 10 points of which 7 are collinear
d. The number of ways of selecting ten balls from the unlimited set of red, green, white and yellow balls,if atleast one ball of each color must be included in the 10 balls selections.

• A. c, d, a, b
• B. c, a, d, b
• C. c, d, b, a
• D. a, d, b, c

Answer 1

Answer: (C)

c, d, b, a

A) No. of diagonals ${=}^{10}{C}_2-10=35$
B) $2940=2^2.3^1.5^1.7^2$
Thus, no. of factors = $(2+1).(1+1).(1+1).(2+1)=36$
C) No. of circles = ${}^{10}{C}_3{-}^7{C}_3=120-35=85$ (since 3 points are required to define a circle)
D) We first select 1 of each color. Now, we need to select 6 more using the 4 types of balls. Number of ways of doing this = ${}^{6+4-1}{C}_{4-1}{=}_3^{9}C=84$
Hence, we have $\left(C\right)>\left(D\right)>\left(B\right)>\left(A\right)$
Hence, (c) is correct.