Each of four friends orders a sweatshirt from a catalog. There are 16 colors of sweatshirts, 7 of which are all cotton and 9 of which are a blend. Each one orders a different color (no repeats) at random. What is the probability that the friends order only cotton sweatshirts?
First, we need to determine the total number of ways the friends can order the sweatshirts. Since each friend orders a different color, there are 16 choices for the first friend, 15 choices for the second friend, 14 choices for the third friend, and 13 choices for the fourth friend. Therefore, the total number of ways the friends can order the sweatshirts is $16 \times 15 \times 14 \times 13 = 43,\!680$.
Next, we need to determine the number of ways the friends can order only cotton sweatshirts. Since there are 7 cotton colors, there are 7 choices for each friend. Therefore, the number of ways the friends can order only cotton sweatshirts is $7 \times 7 \times 7 \times 7 = 2,\!401$.
Finally, we can calculate the probability that the friends order only cotton sweatshirts by dividing the number of ways they can order only cotton sweatshirts by the total number of ways they can order the sweatshirts. The probability is $\dfrac{2,\!401}{43,\!680} = \boxed{\dfrac{53}{960}}$.
Each of four friends orders a sweatshirt from a catalog. There are 16 colors of sweatshirts, 7 of which are all cotton and 9 of which are a blend. Each one orders a different color (no repeats) at random. What is the probability that the friends order only cotton sweatshirts?
A. 1/35
B. 9/455
C. 44/7
D. 1/52
First, we need to determine the total number of ways the friends can order the sweatshirts. Since each friend orders a different color, there are 16 choices for the first friend, 15 choices for the second friend, 14 choices for the third friend, and 13 choices for the fourth friend. Therefore, the total number of ways the friends can order the sweatshirts is $16 \times 15 \times 14 \times 13 = 43,\!680$.
Next, we need to determine the number of ways the friends can order only cotton sweatshirts. Since there are 7 cotton colors, there are 7 choices for each friend. Therefore, the number of ways the friends can order only cotton sweatshirts is $7 \times 6 \times 5 \times 4 = 840$.
Finally, we can calculate the probability that the friends order only cotton sweatshirts by dividing the number of ways they can order only cotton sweatshirts by the total number of ways they can order the sweatshirts. The probability is $\dfrac{840}{43,\!680} = \boxed{\dfrac{9}{455}}$.
Therefore, the correct answer is $\boxed{\text{(B)}\ 9/455}$.