| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Arrangements with grouped categories |
| Difficulty | Standard +0.3 This is a standard permutations question with grouped arrangements and restrictions. Part (i) requires treating groups as single units with repeated letters (two Es, two Ns), part (ii) uses complementary counting, and part (iii) involves selection with constraints. All techniques are routine for S1 level with no novel insight required, making it slightly easier than average. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| e.g. (OAEE)(CPNHGN) or cv | M1 | \(4!/2!\) or \(6!/2!\) seen anywhere |
| \(\dfrac{4!}{2!} \times \dfrac{6!}{2!} \times 2 = 8640\) | M1 A1 [3] | All multiplied by 2 oe |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Total ways \(= 10!/2!2! = 907200\) | B1 | Total ways together correct |
| EE together in \(9!/2!\) ways \(= 181440\) | M1 | EE together attempt alone |
| EE not together \(= 907200 - 181440 = 725760\) | M1 A1 [4] | Considering total \(-\) EE together |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(C_\uparrow\text{P N H G N O A}\) in \(8!/2!\) ways | B1 | \(8!/2!\) seen |
| Insert E in 9 ways | M1 | Interspersing an E, \(\times n\) where \(n = 7, 8, 9\); condone additional factors |
| Insert 2nd E in 8 ways, \(\div 2\) | M1 | Mult by \(9 \times 8 (\div 2)\), \(^9C_2\) or \(^9P_2\) only oe |
| Total \(= 8!/2! \times 9 \times 8 \div 2 = 725760\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| EN** in \(^6C_2\) ways | M1 | \(^6C_x\) or \(^3C_2\) seen alone or mult by \(k > 1\), \(x < 6\), \(y > 2\) |
| M1 | \((1 \times 1 \times)\ ^6C_2\) seen strictly alone or added to their EENN only | |
| \(= 15\) different ways | A1 | |
| EENN in 1 way | B1 | |
| Total 16 ways | A1 [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Listing with at least 8 different correct options | M1 | Value stated or implied by final answer |
| Listing all correct options | M1 | |
| Total \(= 15\) different ways | A1 | correct value stated |
| EENN in 1 way | B1 | |
| Total 16 ways | A1 | Award 16 SRB2 if no method is present |
## Question 6:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| e.g. (OAEE)(CPNHGN) or cv | M1 | $4!/2!$ or $6!/2!$ seen anywhere |
| $\dfrac{4!}{2!} \times \dfrac{6!}{2!} \times 2 = 8640$ | M1 A1 [3] | All multiplied by 2 oe |
### Part (ii):
**First Method:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Total ways $= 10!/2!2! = 907200$ | B1 | Total ways together correct |
| EE together in $9!/2!$ ways $= 181440$ | M1 | EE together attempt alone |
| EE not together $= 907200 - 181440 = 725760$ | M1 A1 [4] | Considering total $-$ EE together |
**Second Method:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $C_\uparrow\text{P N H G N O A}$ in $8!/2!$ ways | B1 | $8!/2!$ seen |
| Insert E in 9 ways | M1 | Interspersing an E, $\times n$ where $n = 7, 8, 9$; condone additional factors |
| Insert 2nd E in 8 ways, $\div 2$ | M1 | Mult by $9 \times 8 (\div 2)$, $^9C_2$ or $^9P_2$ only oe |
| Total $= 8!/2! \times 9 \times 8 \div 2 = 725760$ | A1 | |
### Part (iii):
**First Method:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| EN** in $^6C_2$ ways | M1 | $^6C_x$ or $^3C_2$ seen alone or mult by $k > 1$, $x < 6$, $y > 2$ |
| | M1 | $(1 \times 1 \times)\ ^6C_2$ seen strictly alone or added to their EENN only |
| $= 15$ different ways | A1 | |
| EENN in 1 way | B1 | |
| Total 16 ways | A1 [5] | |
**Second Method:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Listing with at least 8 different correct options | M1 | Value stated or implied by final answer |
| Listing all correct options | M1 | |
| Total $= 15$ different ways | A1 | correct value stated |
| EENN in 1 way | B1 | |
| Total 16 ways | A1 | Award 16 SRB2 if no method is present |6 Find the number of ways all 10 letters of the word COPENHAGEN can be arranged so that\\
(i) the vowels ( $\mathrm { A } , \mathrm { E } , \mathrm { O }$ ) are together and the consonants ( $\mathrm { C } , \mathrm { G } , \mathrm { H } , \mathrm { N } , \mathrm { P }$ ) are together, [3]\\
(ii) the Es are not next to each other.
Four letters are selected from the 10 letters of the word COPENHAGEN.\\
(iii) Find the number of different selections if the four letters must contain the same number of Es and Ns with at least one of each.
\hfill \mbox{\textit{CAIE S1 2016 Q6 [12]}}