I see you've uploaded two images with mathematical problems. Let me transcribe and solve the questions for you.

Questions from the Images:

1) Books Arrangement

There are 4 different Mathematics books and 5 different Science books. In how many ways can the books be arranged on a shelf if: a. There are no restrictions?
b. Books of the same subject must be placed together?
c. They must be placed alternately?

2) Distinct Permutations

Find the number of distinguishable permutations of the digits of the number 348838.

3) 4-Digit Numbers

How many 4-digit numbers can be formed from the digits 1, 3, 5, 6, 8, and 9 if no repetition is allowed?

4) Drawing Prizes

In a general assembly, three major prizes are at stake. In how many ways can the first, second, and third prizes be drawn from a box containing 120 names?

5) Seating Arrangement

There are 12 people in a dinner gathering. In how many ways can the host (one of the 12) arrange his guests around a dining table if: a. They can sit on any of the chairs?

Would you like a detailed solution for each question? Let me know!