Find the number of ways in which 3 numbers in A.P. can be selected from 1,2,3,. ......21.

Ifa, b, c are in A.P then 2b =a +c 
$\therefore$a +c is even. $\therefore$ both a and c are either even or odd. From 1 to 21 there are 10 even and 11 odd natural numbers Therefore the number of ways to select 3 numbers a, b, c in A.P. from 1 to 21 The number of ways to select 2 numbers a, c which are either both even or both odd
${=}^{10}{C}_2{+}^{11}{C}_2=45+55=100$