| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Arrangements with grouped categories |
| Difficulty | Standard +0.3 This is a multi-part permutations question with repeated letters requiring standard techniques (fixing positions, treating groups as single units, alternating patterns, and combinations with restrictions). While it has multiple parts and requires careful counting of repeated letters (3 Gs, 2 As, 2 Es), each part uses routine A-level methods without requiring novel insight or complex problem-solving beyond textbook exercises. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{8!}{3!2!2!} = 1680\) | M1 | 8! Divided by at least one of 3!2!2! oe |
| A1 2 | Correct answer | |
| (ii) \(\frac{5!}{} = 120\) | M1 | 5! Seen (not added, may be divided/multiplied) |
| A1 2 | Correct answer | |
| (iii) \(\frac{5!4!}{3!2!2!} = 120\) | B1 | 5! Or 4! Seen in sum or product in numerator (denominator may by 1) |
| M1 | \(\frac{5!4!}{3!2!2!}\) in a numerical expression | |
| A1 3 | Correct final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Total = 12 ways | M1 | Summing 2 options (could be lists) |
| A1 | 1 correct option | |
| A1 3 | Correct answer |
(i) $\frac{8!}{3!2!2!} = 1680$ | M1 | 8! Divided by at least one of 3!2!2! oe
| A1 2 | Correct answer
(ii) $\frac{5!}{} = 120$ | M1 | 5! Seen (not added, may be divided/multiplied)
| A1 2 | Correct answer
(iii) $\frac{5!4!}{3!2!2!} = 120$ | B1 | 5! Or 4! Seen in sum or product in numerator (denominator may by 1)
| M1 | $\frac{5!4!}{3!2!2!}$ in a numerical expression
| A1 3 | Correct final answer
(iv) GG with AA, AE, EE, RA, RE, RT, TA, TE, = 8 ways
GGG with A, E, R, T = 4 ways
Total = 12 ways | M1 | Summing 2 options (could be lists)
| A1 | 1 correct option
| A1 3 | Correct answer6 (i) Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if the first letter is $R$.\\
(ii) Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if the 3 letters G are together, both letters A are together and both letters E are together.\\
(iii) The letters G, R and T are consonants and the letters A and E are vowels. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if consonants and vowels occur alternately.\\
(iv) Find the number of different selections of 4 letters of the word AGGREGATE which contain exactly 2 Gs or exactly 3 Gs.
\hfill \mbox{\textit{CAIE S1 2013 Q6 [10]}}