You are forming groups where order doesn’t matter, so use the combination formula. If you use 5 ingredients, then there are: \(\frac{10}{5}\frac{9}{4}\frac{8}{3}\frac{7}{2}\frac{6}{1}\)=252 different combinations. If you use 6 ingredients there are \(\frac{10}{6}\frac{9}{5}\frac{8}{4}\frac{7}{3}\frac{6}{2}\frac{5}{1}\)=210 combinations, if you use 7 there are \(\frac{10}{6}\frac{9}{5}\frac{8}{4}\frac{7}{3}\frac{6}{2}\frac{5}{1}\)=120, and if you use 8 there are \(\frac{10}{8}\frac{9}{7}\frac{8}{6}\frac{7}{5}\frac{6}{4}\frac{5}{3}\frac{4}{2}\frac{3}{1}\)=45.Thus, Quantity A is 8, and Quantity B is 5, choice (A).

Start by finding out how many groups of three seniors can be chosen from the five seniors: \(\frac{5}{3}\frac{4}{2}\frac{3}{1}\)=10. Next, multiply that total by the number of individual juniors with which those groups can be paired (7) to form the full committee: 10 × 7 = 70. Quantity B is greater.

60

When calculating the number of games, order does not matter.There are two students in each game, so two slots:\(\frac{6}{2}\frac{5}{1}\), because order does not matter you will divide by the factoral for 15 combinations.Each student plays 4 games against each of the other students, so 4(15) = 60 games are played.

To find the number of different committees. Make your slots and divide by the factoral \(\frac{9}{3}\frac{8}{2}\frac{7}{1}=84\); eliminate choice (B).To find the number of ways to arrange the committee members, just multiply 9 × 8 × 7 = 504, and eliminate partial choice (E).The difference is 504 − 84 = 420, choice (D).

Dividing 100,000 seconds by 3,600 seconds per hour, you get 27 hours plus ⁷⁄₉ hr. Multiplying ⁷⁄₉ hr by 60 minutes per hour, you get 46(²⁄₃) minutes.Therefore, to the nearest minute, 100,000 seconds is equal to 27 hours, 47 minutes. After 24 hours, the time will be 9:30 p.m. Wednesday; 3 hours, 47 minutes after that, it will be 1:17 a.m.,Thursday, choice (D).

First find the number of groups of fish he can select.This is your number of slots: _ _ _ _ _.There are six fish he can choose for the first slot, 5 for the second and so on: 6 5 4 3. Since order doesn’t matter, you need to divide by the factorial of the number of slots:\(\frac{6213}{4321}\). Reduce your number to get 3 × 5 = 15. For plants you have two slots so ³⁄₂ × ²⁄₁ = 3. 3 × 15 = 45.The answer is choice (D).

Substitute the given values into the groups equation:Total = Group 1 + Group 2 − Both + Neither. So 190 = 95 + 75 – 12 + N. N = 32. If you pick a number larger than 12 to represent the number of students who buy both cookies, the number that buy neither cookie also increases.The question asks for the least number that bought neither cookie, so the answer is choice (D).

468,000

List the number of possible options for each character in the password.There are 10 possibilities for the first digit, 9 left for the second, and 8 left for the third.There are 26 possibilities for the first letter and 25 for the second.There are 10 × 9 × 8 × 26 × 25 = 468,000 possible passwords.

Quantity A places a restriction on which meals a chef can cook because each chef must cook a distinct meal. In this case, there would be 3 × 2 × 1 = 6 different assignments. Quantity B does not place a restriction on which meals a chef can cook. In this case, there would be 3 × 3 × 3 = 27 different assignments, choice (B).

Substitute the given values into the groups equation:Total = Group 1 + Group 2 − Both + Neither. 100 = 60 + 40 − 30 + N, giving you N = 30, choice (C).