Write a full solution.
1.1) Find the number of ways that the word GIGGLING can be arranged such that the words can't start with vowels.
1.2) There're 50 numbers from 1 to 50. How many different ways can it be arranged in straight line such that odd numbers are arranged in increasing order and 2,4,6,8 must be next to each other (can be any order).
2.) (I feel like there's a problem in this question) Let be the number of ways of setting people sit on identical round tables such that each tables have at least people. Prove that
for all natural numbers and
3.) Let be natural numbers, prove these by combinatorial proof.
3.2) is an integer.
4.) There're different gifts, give them to of the students (not necessary every gifts are given). If each students get more than gifts, find the number of ways to do that.
5.) There're people in a group (Brilli is one of them). Choose at least people and have them stand on a straight line. The rest of people, except Brilli, are sitting on a circular table. Find the number of ways to do that.
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