1.

- How many seven digit numbers can be formed from the digits 1,1,2,2,3,3,3?
- How many nine digit numbers can be formed from the digits 1,1,2,2,3,3,3,3,3?
- How many six digit numbers can be formed from the digits 0,1,2,3,3,3?
- How many eight digit numbers can be formed from the digits 0,1,1,2,2,3,3,3?

2.
How many integer solutions do the following equations have?

- \(w + x + y + z = 12,\) where \(w \geq 2 , x\geq 2 , y\geq 2 , z \geq 2;\)
- \(w + x + y + z = 10,\) where \(w \geq 0, x\geq 1, y\geq 2, z \geq 3;\)
- \(v + w + x + y + z = 15,\) where \(v \geq 1, w \geq 0, x\geq 1, y\geq 3, z \geq 3.\)

3. Use the Euclidean algorithm to find \(\gcd(a,b)\) and compute integers \(x\) and \(y\) for which
\[
ax+by=\gcd(a,b):
\]
(I) \(a=1717, b=1479\),
(II) \(a=2323, b=2001.\)

4. Determine all non-negative integral solutions of the equation
\[
13x+19y=234.
\]

5. Provide a bound \(N\) for \(n\) such that the equation is solvable in non-negative integers if \(n\geq N:\)
\[
7x+17y=n.
\]

6. Expand the following expression using the binomial theorem:
\[
\left(\frac{x}{3}-2\right)^3.
\]

7. By using mathematical induction, show that \(4^n + 15n - 1\) is divisible by 9 for all \(n \geq 1.\)

8. Prove that \(n^3+2n\) is divisible by 3 for all positive integers \(n.\)

9. Prove by induction that
\[2^n \lt n!\]
for any positive integer \(n\geq 4\).

10. Prove that
\[
\sum_{k=1}^n\frac{1}{(3k-1)(3k+2)}=\frac{n}{6n+4}
\]
for every positive integer \(n\).

11. Prove that
\[
\sum_{k=1}^n\frac{1}{k(k+1)(k+2)}=\frac{n(n+3)}{4(n+1)(n+2)}
\]
for all positive integer \(n\).

12. Let \(a_1=1, a_2=6\) and \(a_{n+1}=7a_n-12a_{n-1},n\geq 2.\) Prove that
\[
a_n=3\cdot 4^{n-1}-2\cdot 3^{n-1}
\]
for all positive integer \(n\).

13. Consider the sequence of real numbers defined by the relations \[x_1=1\]
and \[x_{n+1}=\sqrt{1 + 2x_n}\] for \(n\geq 1.\)
Prove that \(x_n\lt 4\) for all \(n\geq 1.\)

14. Prove that
\[
10^{n}-3^n
\]
is a multiple of 7 for every positive integer \(n\).

15. Prove that
\[
\prod_{k=n+1}^{2n}k=2^n\prod_{k=1}^n (2k-1)
\]
for all positive integer \(n.\)

In what follows \(F_n\) denotes the Fibonacci sequence, which is defined by
\[
F_0=0,\quad
F_1=1,\quad
F_n=F_{n-1}+F_{n-2},\quad n\geq 2.
\]

16. Prove by induction that \[\sum_{i=0}^{n}F_i=F_{n+2}-1.\]

17. Prove by induction that \[\sum_{i=0}^{n-1}F_{2i+1}=F_{2n}.\]

18. Prove by induction that \[\sum_{i=1}^{n}F_{2i}=F_{2n+1}-1.\]

19. Prove by induction that \[\sum_{i=1}^{n}F_i^2=F_{n}F_{n+1}.\]

20. Prove that \[F_n^2-F_{n+1}F_{n-1}=(-1)^{n-1}.\]

21. Prove by induction the following divisibility properties of the Fibonacci numbers.

- 2 divides \(F_{3n},\)
- 3 divides \(F_{4n},\)
- 5 divides \(F_{5n}.\)