Comprehensive assessment covering counting principles, permutations, combinations, Pascal's triangle, binomial theorem, inclusion-exclusion, and the pigeonhole principle.
1. A code consists of 2 letters followed by 4 digits. How many codes are possible if repetition is allowed? (2 marks)
2. In how many ways can the letters of TRIANGLE be arranged? (1 mark)
3. Evaluate $^7P_3$. (1 mark)
4. How many committees of 3 can be chosen from 10 people? (1 mark)
5. In how many ways can 5 people be seated around a circular table? (1 mark)
6. Expand $(x+1)^5$. (2 marks)
7. Find the coefficient of $x^3$ in $(2x+1)^4$. (2 marks)
8. Find the sum of all entries in row 7 of Pascal's triangle. (1 mark)
9. A bag has 20 red, 15 blue, and 10 green marbles. How many must be drawn to guarantee at least 5 of one colour? (2 marks)
10. Prove that $^nC_0 + \\,^nC_1 + \\cdots + \\,^nC_n = 2^n$. (2 marks)