Counting — Practice Problems and Worked Solutions

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17 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Count the number of ordered arrangements.

   $$P(5,2)$$

   Answer: $20$

   Step-by-step solution

   1. $$P(5,2)=\frac{5!}{(5-2)!}$$
   2. $$=20$$
2. 2.

   Count the number of possible selections.

   $$C(6,3)$$

   Answer: $20$

   Step-by-step solution

   1. $$C(6,3)=\frac{6!}{3!(6-3)!}$$
   2. $$=20$$
3. 3.

   Count the number of possible selections.

   $$C(7,2)$$

   Answer: $21$

   Step-by-step solution

   1. $$C(7,2)=\frac{7!}{2!(7-2)!}$$
   2. $$=21$$
4. 4.

   Count the number of ordered arrangements.

   $$P(8,3)$$

   Answer: $336$

   Step-by-step solution

   1. $$P(8,3)=\frac{8!}{(8-3)!}$$
   2. $$=336$$
5. 5.

   Count the number of possible selections.

   $$C(10,4)$$

   Answer: $210$

   Step-by-step solution

   1. $$C(10,4)=\frac{10!}{4!(10-4)!}$$
   2. $$=210$$
6. 6.

   Count the number of ordered arrangements.

   $$P(9,5)$$

   Answer: $15120$

   Step-by-step solution

   1. $$P(9,5)=\frac{9!}{(9-5)!}$$
   2. $$=15120$$
7. 7.

   Count the number of ordered arrangements.

   $$P(6,2)$$

   Answer: $30$

   Step-by-step solution

   1. $$P(6,2)=\frac{6!}{(6-2)!}$$
   2. $$=30$$
8. 8.

   Count the number of possible selections.

   $$C(7,3)$$

   Answer: $35$

   Step-by-step solution

   1. $$C(7,3)=\frac{7!}{3!(7-3)!}$$
   2. $$=35$$
9. 9.

   Count the number of possible selections.

   $$C(8,2)$$

   Answer: $28$

   Step-by-step solution

   1. $$C(8,2)=\frac{8!}{2!(8-2)!}$$
   2. $$=28$$
10. 10.

    Count the number of ordered arrangements.

    $$P(9,3)$$

    Answer: $504$

    Step-by-step solution

    1. $$P(9,3)=\frac{9!}{(9-3)!}$$
    2. $$=504$$
11. 11.

    Count the number of possible selections.

    $$C(12,4)$$

    Answer: $495$

    Step-by-step solution

    1. $$C(12,4)=\frac{12!}{4!(12-4)!}$$
    2. $$=495$$
12. 12.

    Count the number of ordered arrangements.

    $$P(10,3)$$

    Answer: $720$

    Step-by-step solution

    1. $$P(10,3)=\frac{10!}{(10-3)!}$$
    2. $$=720$$
13. 13.

    Count the number of possible selections.

    $$C(11,2)$$

    Answer: $55$

    Step-by-step solution

    1. $$C(11,2)=\frac{11!}{2!(11-2)!}$$
    2. $$=55$$
14. 14.

    Count the number of ordered arrangements.

    $$P(6,4)$$

    Answer: $360$

    Step-by-step solution

    1. $$P(6,4)=\frac{6!}{(6-4)!}$$
    2. $$=360$$
15. 15.

    Count the number of possible selections.

    $$C(9,2)$$

    Answer: $36$

    Step-by-step solution

    1. $$C(9,2)=\frac{9!}{2!(9-2)!}$$
    2. $$=36$$
16. 16.

    Count the number of ordered arrangements.

    $$P(7,4)$$

    Answer: $840$

    Step-by-step solution

    1. $$P(7,4)=\frac{7!}{(7-4)!}$$
    2. $$=840$$
17. 17.

    Count the number of possible selections.

    $$C(12,5)$$

    Answer: $792$

    Step-by-step solution

    1. $$C(12,5)=\frac{12!}{5!(12-5)!}$$
    2. $$=792$$

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