| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Multi-stage selection problems |
| Difficulty | Challenging +1.2 This is a multi-part combinations problem requiring systematic case-by-case analysis for part (i) with constraints, standard permutations with identical objects for part (ii), and careful case consideration for part (iii). While it requires organization and multiple techniques, the individual steps are standard A-level methods without requiring novel insight—harder than average due to the case work and potential for errors, but well within typical Further Maths S1 scope. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Total = 148176 (148000) | M1 | Multiplying three combinations together |
| M1 | Summing more than one sensible option | |
| A1 | Two options correct unsimplified | |
| A1 [4] | Correct final answer | |
| (ii) \(\frac{11!}{5!4!2!} = 6930\) | B1 [1] | Correct answer evaluated |
| Answer | Marks | Guidance |
|---|---|---|
| Total = 6930 | M1 | Summing three options |
| B1 | One option correct | |
| A1 [3] | Correct final answer |
**(i)** Options 5 bat 5 bl 1 Wk in ${}^{10}C_5 \times {}^9C_5 \times {}^7C_1 = 63504$ ways
or 5 bat 4 bl 2 Wk in ${}^{10}C_5 \times {}^9C_4 \times {}^7C_2 = 31752$ ways
or 6 bat 4 bl 1 Wk in ${}^{10}C_6 \times {}^9C_4 \times {}^7C_1 = 52920$ ways
Total = 148176 (148000) | M1 | Multiplying three combinations together
| M1 | Summing more than one sensible option
| A1 | Two options correct unsimplified
| A1 [4] | Correct final answer
**(ii)** $\frac{11!}{5!4!2!} = 6930$ | B1 [1] | Correct answer evaluated
**(iii)** Omit a pen $\frac{10!}{4!4!2!} = 3150$
Omit a diary $\frac{10!}{5!3!2!} = 2520$
Omit a notebook $\frac{10!}{5!4!} = 1260$
Total = 6930 | M1 | Summing three options
| B1 | One option correct
| A1 [3] | Correct final answer4 A cricket team of 11 players is to be chosen from 21 players consisting of 10 batsmen, 9 bowlers and 2 wicketkeepers. The team must include at least 5 batsmen, at least 4 bowlers and at least 1 wicketkeeper.\\
(i) Find the number of different ways in which the team can be chosen.
Each player in the team is given a present. The presents consist of 5 identical pens, 4 identical diaries and 2 identical notebooks.\\
(ii) Find the number of different arrangements of the presents if they are all displayed in a row.\\
(iii) 10 of these 11 presents are chosen and arranged in a row. Find the number of different arrangements that are possible.
\hfill \mbox{\textit{CAIE S1 2011 Q4 [8]}}