Permutations and Combinations.

1. How many numbers less than `n!` are divisible by all prime numbers less than `n`? What is the number for `1000`? Please show the work-out.

2. Prove that `2nCn` is always divisible by `2^{2}`, except when n is a power of 2.

3. Prove that `2^{n+1}C2^{n}` is divisible by 2, but not by 4, for all positive integral values of `n`.

4. Prove that, if `p` is a prime, then for any non-negative number `a`, `(a^p = a) mod p`.

5. Choose integer dimensioned rectangles (both width and height), so that, width `<= 2n` and height `<= 2n`. What is the probability that the resulting rectangle's areas is less than `n^{2}`.

a. If you were to bet on whether the resulting rectangles area will be less than `n^{2}` or not, what would you bet on?

6. Consider the quadratic equation `ax^2 + bx + c = 0`, with integer (not necessarily positive) co-efficients `a`, `b` and `c`, such that all of them are less than or equal to 100, in absolute value. How many numbers from 1 to 100 are roots of such an equation with some `a`, `b` and `c` combination?

7. Define a nice number as a rational number, with numerator and denominator positive and not exceeding some number. indicates the number of nice numbers, in , What is the formula for N?.

8. Find the number for nice numbers (question 3), when numerator can exceed `x` and denominator can exceed `y`, but they should be reducible to something where they do not exceed `x` and `y`, respectively. If a formula is not possible, give an explanation why?

9. How many 4 digit numbers are there such that, the minimal positive difference between their digits is 1. "minimal" means, if the number is `abcd`, then taking any combination like, `a + b + c - d` or `a - b - c + d` that results in a positive number, can one get 1? (Use only plus and minus).

10. What is the minimum length (lower bound) of a string of digits, that contains all possible permutations of numbers 1 to 9?

a. Prove that such a string will be not be more than `9 * 9!` in length.

a. Prove that such a string will be not be more than `9 * 9!` in length.

11. The distance between `2` permutations of numbers `1` to `n`, is defined as the least number of position flips, to make one permutation same as the other. For example, `123` is at a distance of `2` to `231`, because it can be got by first flipping `1` and `2` in `123`, getting `213`, then again flipping `1` and `3`, getting `231`

a. What is the minimum distance between two distinct permutations from `1` to `n`.

b. What is the maximum distance between two distinct permutations from `1` to `n`.

c. What is the sum of all pairwise distances of all permutations of `n` elements, from `1` to `n`. (Distance between `p_{a}` and `p_{b}` is counted only once, not twice).

a. What is the minimum distance between two distinct permutations from `1` to `n`.

b. What is the maximum distance between two distinct permutations from `1` to `n`.

c. What is the sum of all pairwise distances of all permutations of `n` elements, from `1` to `n`. (Distance between `p_{a}` and `p_{b}` is counted only once, not twice).

12. In a `2m` page book, some sheets are missing. In how many ways, some `n` sheets that may be missing from the `2m` page book. (A sheet has `2` pages, no cover for the book).

13. `10000! = (100!)^{k} * p`, for some positive numbers k, p. Determine the maximum value for k.

14. Find the limit of `1/1! + 1/(1! + 2!) + 1/(1! +2! + 3!) + ...`, when the number of terms in denominator tends to infinity.

13. `10000! = (100!)^{k} * p`, for some positive numbers k, p. Determine the maximum value for k.

14. Find the limit of `1/1! + 1/(1! + 2!) + 1/(1! +2! + 3!) + ...`, when the number of terms in denominator tends to infinity.

15. Prove that `1! + 2! + 3! + ... + n! < 2 * \sqrt{n} * ((n + e) / e)^n` for positive n.

16. Prove that `1! + 2! + 3! + ... + n! > (e + 1)/sqrt_{2 \pi} * (n / e)^n` for positive n.

17. Prove that `(6n + 4) C (3n + 2)` is always divisible by 3.

16. Prove that `1! + 2! + 3! + ... + n! > (e + 1)/sqrt_{2 \pi} * (n / e)^n` for positive n.

17. Prove that `(6n + 4) C (3n + 2)` is always divisible by 3.

18. Consider a quadrilateral to be a pythagorean, if all its sides are integers, and diagonal measures are also integers. Find the number of pythagorean quadrilaterals with dimensions (`<= N`).

19. Prove that there are infinitely many values of `n` for which `2nCn` is not divisible by a given odd prime number `p`.

20. Prove that only every 3rd number in fibonacci sequence is divisible by 2, only every 4th is divisible by 3, and every 5th by 5, and every sixth number is divisible by 8. What are the numbers for the lucas sequence? Prove them.

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