| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Arrangements with adjacency requirements |
| Difficulty | Moderate -0.8 This is a straightforward permutations question with standard techniques: (i) arrangements with repeated letters using n!/r!, (ii) treating adjacent letters as a single unit, and (iii) basic probability with 'at least one' using complement or direct counting. All are textbook exercises requiring only recall of standard methods with minimal problem-solving. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| 3(i) | \(\frac{7!}{3! \times 2(!)} = 420\) | M1M1: —; A1: 3 |
| 3(ii) | \(\frac{5!}{2(!)} = 60\) | M1: —; A1: 2 |
| 3(iii) | \(1 - ^1/_7 \times ^6_{10}\) or \(1 - ^{\text{C}}_{2}/^{\text{C}}_2\) or \(1 - ^{\text{P}}_2/^{\text{P}}_2\) or \(^1/_6 \times ^{\text{C}}_2 + ^1/_6 \text{ or } ^{\text{C}}_1/^{\text{C}}_2 + ^{\text{C}}_{\text{C}}1/^{\text{C}}_2\) | M1M1: — |
| \(= ^5/_7\) or \(0.714\) (3 sfs) | A1: 3 |
3(i) | $\frac{7!}{3! \times 2(!)} = 420$ | M1M1: —; A1: 3 | M1: $7!/(a \text{ factorial})$ or $\ldots ÷ (3! \times 2(!))$; M1: all correct |
3(ii) | $\frac{5!}{2(!)} = 60$ | M1: —; A1: 2 | M1: $5!$ seen (not part of C) or $5 \times 4!$ or 120 seen or $\ldots ÷ 2(!)$ alone |
3(iii) | $1 - ^1/_7 \times ^6_{10}$ or $1 - ^{\text{C}}_{2}/^{\text{C}}_2$ or $1 - ^{\text{P}}_2/^{\text{P}}_2$ or $^1/_6 \times ^{\text{C}}_2 + ^1/_6 \text{ or } ^{\text{C}}_1/^{\text{C}}_2 + ^{\text{C}}_{\text{C}}1/^{\text{C}}_2$ | M1M1: — | M1: $1 - \text{ prod }$ or $1 - \ldots/^{\text{C}}_2$ or $1 - ^{\text{C}}_1 \ldots$ (or Ps) or add 3 prods or add 2 correct prods or $^{\text{C}}_1/^{\text{C}}_2$ or $^{\text{C}}_{\text{C}}1/^{\text{C}}_2$ or add $> 5$ out of 7 correct prods; M1: all correct |
| $= ^5/_7$ or $0.714$ (3 sfs) | A1: 3 | |
**Total: 8**
---3 Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a row.\\
(i) How many different arrangements of the letters are possible?\\
(ii) In how many of these arrangements are all three Ds together?
The 7 cards are now shuffled and 2 cards are selected at random, without replacement.\\
(iii) Find the probability that at least one of these 2 cards has D printed on it.
\hfill \mbox{\textit{OCR S1 2006 Q3 [8]}}