Arrange all of them in a row. Consider the first person, who can be paired with by people. The second person will then have choices. Repeat this process, and the ways of ordering is therefore . Simplifying gives

2. The Angle Between Lines

The angle makes with the -axis satisfies . Similarly, the angle makes with the -axis satisfies . With a diagram, we can see that the angle between these two lines is , so

3. An Integral

Consider two integrals and . Then

using the trigonometric identities and integration by parts. This gives .

4. Multiples of 2, 3, and 5

Since 2, 3, and 5 are all prime numbers, removing multiples of one number doesn't affect the proportion of multiples of the others. So the answer is simply

5. A Probability about Coins

The probability he selects the biased coin is while for an unbiased coin, it is . Consider the Bayes' theorem. and . Therefore, .

6. A Packing Case

If the packing case eventually comes to rest, there must be an upward acceleration along the slope. Resolving forces gives and where . That means . Under this condition, the acceleration is by Newton's second law. If the distance it goes before stopping is , by suvat, where . Hence, . The condition ensures that the expression inside the surd is greater than 0.

7. Two Binomial Identities

(i) Consider the set . On the one hand, one may calculate the number of subsets with elements as since one only needs to select elements from of them; on the other hand, every subset of the original set either contains the element 1 or it doesn't, and in the first case, the ways of selection is while in the second, there are choices. Hence, .

(ii) Consider the set . On the one hand, one may calculate the number of subsets with elements as since one only needs to select elements from of them; on the other hand, every subset of the original set could contain both the elements 1 and 2, or one of them, or neither. In the first case, the ways of selection is ; in the second, there are choices; in the third, methods are possible. Hence, .

8. The Rod

Let be the mass of the rod. The loss of GPE is given by and the rotational kinetic energy is given by KE = where . By conservation of energy, these two quantities are equal; consequently, . Hence the speed of a point on the unhinged end of the rod is given by .

9. An Approximation

Observe first that is a continuous function. Now, for ; since is very large, we may assume this holds. Also, from the definition of , . For , , so . Hence, by the Intermediate Value Theorem, such that . Suppose that such a solution is not unique, then let two such distinct solutions be and . Since , . WLOG, let , then where . Since , . But , giving . This leads to a contradiction, which implies .

Now if , i.e. , . As for reasonable sized which occur when is large, we have . Letting , we see that , so and . For large M, we may write , which gives .

10. Twenty Balls

(i) There are ways to pick exactly one red, and ways to pick the remaining non-reds. The probability is therefore .

(ii) We first choose 3 colours out of 4, giving ways. There are ways to pick exactly one colour A, ways to pick exactly one colour B, and ways to pick exactly one colour C. The probability is therefore .

(iii) The conditions is fulfilled if we have 3 blues, 2 blues, or 1 blue and 0 yellows. The probabilities are , , and respectively. Adding them together gives the final result.