Please solve the following problems ( kindly show solutions and explanations) thank you so much

Topic: permutations

I. Eight children join hands and form a circle. In how many ways could the circle be formed?

2. Find the number Of ways in which 5 people A, B, C, D, and E can be seated at a round table, such that A and B always sit together.

3. Nine books are to be arranged on a shelf. There are 3 Math books, 4 Science books, and 2 English books. How many distinct arrangements are possible?

4. Find the number of distinguishable permutations of the digits of the number 287,772.

5. Marvin, Albert, Raff, Jeff and Ronald want to take a photo in which three of them are lined up in a row. How many different photos are possible?

6. How many photographs will 9 friends have if they will take a group picture of 4 at a time?

7. You have been asked to judge an. art contest with 20 entries. In how many ways can you assign 1st, 2nd, and 3rd place?

8. How many words can we make by rearranging the letters of the word " PHILIPPINES"?

9. There are 10 people in a dinner gathering. In how many ways can they sit around a dining table if they can sit on any of the chairs?

Distinct/Distinguishable Permutation

1. How many permutations are there of the

letters in the word "greet"?

2. How many distinct permutations can be

formed from the word "WEDNESDAY'?

3. In how many ways can you arrange the

letters in the word RHOMBUS?

Circular Permutation

4. In how many ways can 4 people be

seated around a circular table?

5. In how many ways can 8 people form a

circle a folk dance?

6. In how many ways can 5 people be

seated around a circular table if two of

them insist on sitting beside each other?