Question

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?

Answer 1

Answer: 4368

(The other answer was wrong)

Call the 5 chairs a, b, c, d, e. Then, since they can't be adjacent, we need at least one chair between them. So it looks something like this

a [] b [] c [] d [] e

We have 15 chairs left (20 - 15 = 5), and we must insert 4, so now there are 15 - 4 = 11 chairs to distribute across six slots:

(here) a (here) b (here) c (here) d (here) e (here)

This is just solving a + b + c + d + e + f = 11 where a, b, c, d, e, f are non-neg ints. This comes out to 16 choose 5, which is 4368.

Also if you need help on AoPS HW use the Message Board first before Brainly, Web2.0calc, MSE, etc plz!

Hope that helped,

-sirswagger21

Answer 2

Answer:

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 .