Question

Thirteen persons are sitting in a row. Number of ways in which four persons can be selected so that no two of them are consecutive is also equal to

• A. Number of ways in which letters of word CINEMA can be arranged if all vowels are together
• B. Number of numbers lying between 100 and 1000 using only the digits 1, 2, 3, 4, 5, 6, 7 without repetition
• C. Number of ways in which 4 alike flowers can be distributed among 10 girls so that each girl gets atmost one
• D. Number of triangles which can be formed by joining 12 points in a plane of which 5 are collinear

Answer 1

Answer: (B), (C), (D)

The correct options are
B

Number of numbers lying between 100 and 1000 using only the digits 1, 2, 3, 4, 5, 6, 7 without repetition

C

Number of ways in which 4 alike flowers can be distributed among 10 girls so that each girl gets atmost one

D

Number of triangles which can be formed by joining 12 points in a plane of which 5 are collinear

Selecting 4 people out of 13 people sitting in a row such that no two consecutive people are selected is the scenario same as choosing 4 seats out of 13 empty seats such that no two seats chosen are together.

We start with 9 empty seats.

So, now there are 10 places for the choosing those 4 seats which can be done in ${}^{10}{C}_4=210$ ways.

(A) CINEMA $C\begin{array}{c}IEA\end{array}NM$

Can be arranged in 4! $\times$ 3! = 24 $\times$ 6 = 144

(B) ___ ___ ___ Three digit numbers possible are 7 $\times$ 6 $\times$ 5 = 210

(C) 4 alike flowers can be distributed in ${}^{10}{C}_4$ ways

(D) ${}^{12}{C}_3{-}^5{C}_3=220-10=210{=}^{10}{C}_4$