Comprehensive assessment covering counting principles, permutations, combinations, Pascal's triangle, binomial theorem, inclusion-exclusion, and the pigeonhole principle.
A code has 2 letters followed by 3 digits. Repetition is allowed. How many codes are possible?
The value of $^7P_3$ is:
The value of $^{10}C_3$ is:
How many ways can 6 people be seated around a circular table?
In how many ways can the letters of LEVEL be arranged?
A committee of 4 is chosen from 9 people. Which expression counts the committees?
How many 4-digit numbers can be formed from the digits 1 to 9 if no digit is repeated?
The coefficient of $x^2$ in $(1 + x)^5$ is:
The sum of all entries in row $n$ of Pascal's triangle is:
Pascal's identity states that:
The coefficient of $x^3$ in $(2x + 1)^4$ is:
If there are 13 people, the pigeonhole principle guarantees that at least two people:
For finite sets $A$ and $B$, $|A \cup B|$ equals:
How many ways can 5 books be arranged on a shelf if two particular books must stay together?
Which situation should be counted with permutations rather than combinations?
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)