| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Arrangements with alternating patterns |
| Difficulty | Moderate -0.3 This is a straightforward permutations and combinations question testing standard techniques: treating grouped letters as single units, counting arrangements with restrictions, and basic selection problems. Part (ii) requires careful counting of alternating patterns but follows a mechanical approach. All parts are routine applications of A-level methods with no novel insight required, making it slightly easier than average. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{6!}{2!} = 360\) | B1 | 6! Seen alone |
| B1 | 2 | Dividing by 2! only |
| (ii) \(\frac{4!}{2!} \times \frac{4!}{3!}\) | B1 | 4! seen mult |
| B1 | Dividing by 2! or 3! (Mult by 4 implied B1B1) | |
| \(= 48\) | B1 | 3 |
| (iii) 1N and 1A: N A xx in \(^3C_2\) = 3 ways | M1 | \(^3C_x\) or \(^2C_x\) seen alone |
| A1 | 2 | Correct answer |
| (iv) 0 A : Nxxx = 1 way | M1 | Finding ways with 0 or 2 or 3 As |
| 2 As: NAAx in \(^3C_1\) = 3 ways | M1 | Summing 3 or 4 options |
| 3 As: NAAA in 1 way | ||
| Total = 8 ways | A1 | 3 |
(i) $\frac{6!}{2!} = 360$ | B1 | 6! Seen alone
| B1 | 2 | Dividing by 2! only
(ii) $\frac{4!}{2!} \times \frac{4!}{3!}$ | B1 | 4! seen mult
| B1 | Dividing by 2! or 3! (Mult by 4 implied B1B1)
$= 48$ | B1 | 3 | Correct answer
(iii) 1N and 1A: N A xx in $^3C_2$ = 3 ways | M1 | $^3C_x$ or $^2C_x$ seen alone
| A1 | 2 | Correct answer
(iv) 0 A : Nxxx = 1 way | M1 | Finding ways with 0 or 2 or 3 As
2 As: NAAx in $^3C_1$ = 3 ways | M1 | Summing 3 or 4 options
3 As: NAAA in 1 way | |
Total = 8 ways | A1 | 3 | Correct answer6 Find the number of different ways in which all 8 letters of the word TANZANIA can be arranged so that\\
(i) all the letters A are together,\\
(ii) the first letter is a consonant ( $\mathrm { T } , \mathrm { N } , \mathrm { Z }$ ), the second letter is a vowel ( $\mathrm { A } , \mathrm { I }$ ), the third letter is a consonant, the fourth letter is a vowel, and so on alternately.
4 of the 8 letters of the word TANZANIA are selected. How many possible selections contain\\
(iii) exactly 1 N and 1 A ,\\
(iv) exactly 1 N ?
\hfill \mbox{\textit{CAIE S1 2014 Q6 [10]}}