| Exam Board | OCR |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2017 |
| Session | December |
| Marks | 7 |
| Topic | Permutations & Arrangements |
| Type | Specific items together |
| Difficulty | Moderate -0.3 This is a straightforward permutations and combinations problem with standard techniques. Part (i) uses the 'treat items as a block' method (9!/2! for total arrangements, 8! for E's together), and part (ii) is a basic 'at least' probability using combinations. Both are textbook exercises requiring routine application of formulas with no novel insight, making it slightly easier than average for A-level. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| \(2 \times 9!\) | M1 | 9! seen, with or without 2 |
| \(\div 10!\) | M1 | \(\div 10!\) with or without 2 |
| \(= \frac{1}{5}\) or 0.2 | A1 | Answer, oe but must be exact |
| Answer | Marks | Guidance |
|---|---|---|
| \(^5C_3 \times ^5C_1\) | M1 | \(^5C_3\) with or without \(^5C_1\) |
| \(+ ^5C_4\) | M1 | |
| \(\div ^{10}C_4\) | M1 | Needs at least one M1 |
| \(= \frac{11}{42}\) | A1 | \(\frac{35}{210}\) or \(\frac{11}{42}\) or awrt 0.262, www |
## (i)
$2 \times 9!$ | M1 | 9! seen, with or without 2
$\div 10!$ | M1 | $\div 10!$ with or without 2
$= \frac{1}{5}$ or 0.2 | A1 | Answer, oe but must be exact
**Guidance:** If working insufficient, give B3 for correct answer, otherwise 0
## (ii)
$^5C_3 \times ^5C_1$ | M1 | $^5C_3$ with or without $^5C_1$
$+ ^5C_4$ | M1 |
$\div ^{10}C_4$ | M1 | Needs at least one M1
$= \frac{11}{42}$ | A1 | $\frac{35}{210}$ or $\frac{11}{42}$ or awrt 0.262, www
**Guidance:** Or: $\frac{5}{10} \times \frac{4}{9} \times \frac{3}{8} \times \frac{2}{7} = \frac{1}{42} = 0.0238$ or $\frac{5}{10} \times \frac{4}{9} \times \frac{3}{8} \times \frac{2}{7} = \frac{5}{84} = 0.0595$ or $\times 4$; Total $\frac{11}{42}$ or a.r.t. 0.262
---2 Each letter of the words NEW COURSE is written on a card (including one blank card, representing the space between the words), so that there are 10 cards altogether.\\
(i) All 10 cards are arranged in a random order in a straight line. Find the probability that the two cards containing an E are next to each other.\\
(ii) 4 cards are chosen at random. Find the probability that at least three consonants ( $\mathrm { N } , \mathrm { W } , \mathrm { C } , \mathrm { R } , \mathrm { S }$ ) are on the cards chosen.
\hfill \mbox{\textit{OCR FS1 AS 2017 Q2 [7]}}