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(D) Let $X$ be the maximum number on the $7$ selected coupons. Since the coupons are selected with replacement,each selection is independent. The prob

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(D) Let $X$ be the maximum number on the $7$ selected coupons. Since the coupons are selected with replacement,each selection is independent. The probability that a single selected coupon has a number less than or equal to $9$ is $P(X \le 9) = \frac{9}{15} = \frac{3}{5}$. The probability that a single selected coupon has a number less than or equal to $8$ is $P(X \le 8) = \frac{8}{15}$. The probability that the maximum number is exactly $9$ is given by $P(X = 9) = P(X \le 9) - P(X \le 8)$. $P(X = 9) = (\frac{9}{15})^7 - (\frac{8}{15})^7 = (\frac{3}{5})^7 - (\frac{8}{15})^7$.

$X = x$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$P(X = x)$	$0$	$k$	$2k$	$2k$	$3k$	$k^2$	$2k^2$	$7k^2 + k$

$15$ coupons are numbered from $1$ to $15$. Seven coupons are selected at random with replacement. What is the probability that the maximum number on the selected coupons is $9$?

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