Answer:
### Summary of Answers:
1. FALSE
2. TRUE
3. FALSE
4. TRUE
5. FALSE
6. FALSE
7. TRUE
8. FALSE
9. FALSE
10. TRUE (with clarification needed on the calcula
Step-by-step explanation:
1. In permutation, order does not matter.**
Answer: FALSE
(In permutations, the order does matter.)
2. If objects are arranged in a circle, the number of permutations of the said objects is (n-1)
Answer: TRUE
(When arranging n$ objects in a circle, the number of distinct arrangements is (n-1.)
3. The number of permutations of the word CHOOSE is 500.
Answer: FALSE
(The word "CHOOSE" has repeated letters, and the correct calculation gives a different number.)
4. There are 2,520 different ways the 5 bicycles can be parked if there are 7 available parking spaces.*
Answer: TRUE
(This is a permutation problem where
[tex]P(7, 5) = \frac{7!}{(7-5)!} = 7 \times 6 \times 5 \times 4 \times 3 = 2520.[/tex]
5. There are 300 different ways digits 1, 2, 3, 4, 5, and 6 can be formed if no repetition is allowed.
Answer: FALSE
(The number of arrangements of 6 digits without repetition is 6 = 720.)
6. If objects are arranged in a ring or key chain, the number of permutations of the said objects is
[tex]P = \frac{n!}{r! s! t! h!}[/tex]
Answer: FALSE
(The formula for arrangements in a ring is $(n-1)!for distinct objects, not the one given.)
7. Your friend can arrange 5 books on a shelf in 120 ways.
Answer: TRUE
(The number of arrangements of 5 books is 5! = 120.)
8. The formula for the permutation of n objects taken r at a time is n!
Answer: FALSE
(The correct formula is $P(n, r) = \frac{n!}{(n-r)!}$.)
9. Permutation with repetition or the distinguishable permutation where r objects are alike, s objects are alike, t objects are alike, u objects are alike and so on, has the formula:
[tex]P(n, n) = n! or _{n}P_{n} = n![/tex]
Answer: FALSE
(The formula for distinguishable permutations is
[tex]\frac{n!}{r_1! r_2! \ldots r_k!}[/tex]
where r_i are the counts of indistinguishable objects.)
10. There are 210 ways the 5 couples can arrange themselves in a row where each couple must stay together for their pictures to be taken.*
Answer: TRUE
(If each couple is treated as a single unit, there are 5! arrangements of the couples, and within each couple, there are 2 arrangements, giving 5! x 2⁵ = 120 \times 32 = 3840.
