A lecture hall contains $20$ chairs, all lined in a row. What is the number of ways that five chairs can be chosen, so that no two chairs are adjacent?

Step-by-step explanation:

If we choose chairs having odd number in the row

no of chairs from which selection is made = 10

no of chairs to be selected = 5

no of ways = 10C₅

similarly if we choose hairs having even numbers only,

similar to above, no of ways

= 10C₅

Total no of ways

= 2 x 10C₅

= 2 x 10 x 9 x 8 x 7 x 6 / 5 x 4x3 x 2 x 1

= 504.