Question 7 and 8 Exercise 6.3

Solutions of Question 7 and 8 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

Consider a group of $20$ people. If everyone shakes hands with everyone else, how many handshake takes place?

Since everyone will shake hand with every other person,

therefore the total number of shake hands are: \begin{align}{ }^{20} C_2&=\dfrac{20 !}{(20-2)2!}!\\ &=\dfrac{20!}{18!\cdot 2!}\\ &=190\end{align}

A student is to answer $7$ out of $10$ questions in an examination. How many choices has he, if he must answer the first $3$ questions?

Option to choose different $7$ questions out of $10$ to answer are: $${ }^{10} C_7=\dfrac{10 !}{(10-7) ! 7 !}=120$$ If he must answer the first three,

then the remaining questions are $7$ out of which he has to answer $4.$

In this case the total number of options are to select $4$ are: $${ }^7 C_4=\dfrac{7 !}{(7-4) ! 4 !}=35.$$

Hence he has total $35$ option to select if he must answer the first three questions out of $10.$