When a ticket is bought, there are 2 outcomes. It's either it comes with a popcorn or not. These outcomes are independent. This means that it is a binomial distribution. We would apply the binomial ditribution formula which is expressed as
P(x) = nCx * p^x * q^(n - x)
where
n is the number of trials
x is the number of successess
p is the probability of success
q is the probability of failure
A success in this case is the outcome of buying a ticket that has popcorn coupons. Thus,
p = 0.717
q = 1 - p = 1 - 0.717 = 0.283
n = 21
x = 14
We want to find P(x = 14)
Thus,
P(x = 14) = 21C14 * 0.717^14 * 0.283^(21 - 14)
P(x = 14) = 116280 0* 0.717^14 * 0.2837
P(x = 14) = 0.160
the probabilitythat exactly 14 of the tickets have popcorn coupons is 0.160)