A club of 9 students assigns president, vice president, and secretary, with no student holding two offices. Using ${}_9{P}_3$, how many officer slates are possible?

A robotics team of 8 students chooses a captain, programmer, and driver, and roles cannot repeat. Evaluate ${}_8{P}_3$ for the number of role assignments.

A coach places 6 runners into first, second, third, and fourth relay positions. No runner repeats. Using ${}_6{P}_4$, find the number of lineups.

A teacher assigns 10 students to speaker, timer, recorder, and materials manager, with no student assigned twice. Use ${}_{10}{P}_4$ to count the assignments.

At a science fair, 7 projects can win gold, silver, and bronze, with no ties. Using ${}_7{P}_3$, how many award results are possible?

A band of 12 students names a drum major, assistant, and librarian, and one student cannot hold two roles. Evaluate ${}_{12}{P}_3$.

An art show chooses 5 photos to fill first, second, and third display spots, with no photo repeated. Use ${}_5{P}_3$ to count the displays.

A quiz team of 11 students assigns captain, scorekeeper, spokesperson, and timekeeper, with no repeats. Find ${}_{11}{P}_4$.

A theater group has 6 actors available for lead, understudy, and narrator, with no actor taking two parts. Use ${}_6{P}_3$ to count casts.

A cooking contest ranks 9 finalists as first, second, third, and fourth place, with no ties. Evaluate ${}_9{P}_4$ for the possible rankings.

A library display will feature 4 of 9 novels, and the order on the list does not matter. Using ${}_9{C}_4$, how many groups are possible?

A lab has 8 water samples and selects 3 to retest, with order not considered. Evaluate ${}_8{C}_3$ for the number of sample groups.

Student council chooses a 5-person committee from 12 volunteers, with no offices assigned. Use ${}_{12}{C}_5$ to count the committees.

A gardener chooses 4 of 11 herb varieties for one bed, and arrangement is not part of the choice. Find ${}_{11}{C}_4$.

A coach selects 2 of 10 players for a survey, and the order of names is ignored. Using ${}_{10}{C}_2$, how many pairs are possible?

A bakery chooses 3 of 7 cupcake flavors for a sampler box, with order not counted. Evaluate ${}_7{C}_3$.

A museum selects 6 of 10 posters for a wall group, and their order in the selection does not matter. Find ${}_{10}{C}_6$.

A yearbook editor chooses 3 of 10 candid photos for a page, and order is not part of the choice. Use ${}_{10}{C}_3$.

A farm chooses 4 of 8 seed packets for a test plot, with no order assigned. Evaluate ${}_8{C}_4$ for the possible packet groups.

A clinic selects 2 of 9 appointment days to audit, and the order selected does not matter. Find ${}_9{C}_2$.

Evaluate ${}_7{P}_2$, the number of ordered arrangements of 2 objects chosen from 7 distinct objects.

Find the value of ${}_6{P}_2$, the number of ordered arrangements of 2 objects chosen from 6 distinct objects.

Evaluate ${}_{10}{P}_3$ for ordered arrangements of 3 objects selected from 10 distinct objects.

Evaluate ${}_8{C}_2$, the number of unordered selections of 2 objects chosen from 8 distinct objects.

Find the value of ${}_9{C}_3$ for unordered selections of 3 objects from 9 distinct objects.

Evaluate ${}_{11}{P}_2$, the number of ordered arrangements of 2 objects chosen from 11 distinct objects.

Compute ${}_{12}{C}_2$, the number of unordered pairs chosen from 12 distinct objects.

Evaluate ${}_7{C}_4$, the number of unordered selections of 4 objects from 7 distinct objects.

A raffle drum holds 12 tickets, 5 bought by one family, and 3 tickets are drawn together. Using combinations, find the probability all 3 are that family's, as a fraction in lowest terms.

A bowl has 10 slips, 4 labeled Lab A, and 2 slips are drawn together. Using combinations, find the probability both slips are Lab A, as a fraction in lowest terms.

A drawer contains 15 badges, 6 blue and 9 gray, and 2 badges are chosen at the same time. Use combinations to find the probability both are blue, as a lowest-terms fraction.

A deck has 20 cards, 7 red and 13 black, and 3 cards are drawn together. Using combinations, find the probability all 3 are red, as a fraction in lowest terms.

A box has 14 machine parts, 4 defective and 10 working, and 2 parts are selected together. Use combinations to find the probability both are defective, as a lowest-terms fraction.

A basket holds 11 snacks, 5 nut-free and 6 with nuts, and 2 snacks are chosen together. Using combinations, find the probability both are nut-free, as a fraction in lowest terms.

A playlist has 13 songs, 4 jazz and 9 rock, and 3 songs are selected as an unordered short set. Using combinations, find the probability all are jazz, as a lowest-terms fraction.

A box contains 16 collars, 6 reflective and 10 plain, and 3 collars are selected together. Use combinations to find the probability all 3 are reflective, as a lowest-terms fraction.

An envelope holds 18 stickers, 7 stars and 11 circles, and 2 stickers are drawn together. Using combinations, find the probability both are stars, as a lowest-terms fraction.

A cooler has 9 drinks, 3 lemon and 6 berry, and 2 drinks are chosen together. Use combinations to find the probability both are lemon, as a fraction in lowest terms.

A garden tray has 12 seedlings, 4 basil and 8 tomato, and 3 seedlings are chosen together. Using combinations, find the probability all are basil, as a lowest-terms fraction.

A shelf has 17 books, 5 mysteries and 12 biographies, and 2 books are selected together. Use combinations to find the probability both are mysteries, as a lowest-terms fraction.