CAIE S1 2014 June — Question 2 4 marks

Exam BoardCAIE
ModuleS1 (Statistics 1)
Year2014
SessionJune
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCombinations & Selection
TypeCommittee with gender/category constraints
DifficultyStandard +0.3 This is a straightforward combinations problem requiring students to identify the constraint (at least one from each year), recognize they need to use complementary counting or casework, and apply C(n,r) correctly. The small numbers (11 total members, selecting 5) make calculations manageable, and the method is a standard textbook exercise slightly above routine due to the constraint handling.
Spec5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems
2 A school club has members from 3 different year-groups: Year 1, Year 2 and Year 3. There are 7 members from Year 1, 2 members from Year 2 and 2 members from Year 3. Five members of the club are selected. Find the number of possible selections that include at least one member from each year-group.
Question 2:
AnswerMarks Guidance
WorkingMark Guidance
\(Y1(7)\ Y2(2)\ Y3(2)\): \(1\ \ 2\ \ 2 = 7 \times 1 \times 1 = 7\)B1 One unsimplified correct 3-factor product of combinations
\(2\ \ 1\ \ 2 = ^7C_2 \times ^2C_1 \times 1 = 42\)B1 A second unsimplified correct 3-factor product of combinations
\(2\ \ 2\ \ 1 = ^7C_2 \times 1 \times ^2C_1 = 42\)
\(3\ \ 1\ \ 1 = ^7C_3 \times ^2C_1 \times ^2C_1 = 140\)M1 Summing 3 or 4 options; allow perms, wrong combs but second numbers must sum to 5 etc.
Total \(= 231\)A1 [4] Correct answer 
## Question 2:

| Working | Mark | Guidance |
|---------|------|----------|
| $Y1(7)\ Y2(2)\ Y3(2)$: $1\ \ 2\ \ 2 = 7 \times 1 \times 1 = 7$ | B1 | One unsimplified correct 3-factor product of combinations |
| $2\ \ 1\ \ 2 = ^7C_2 \times ^2C_1 \times 1 = 42$ | B1 | A second unsimplified correct 3-factor product of combinations |
| $2\ \ 2\ \ 1 = ^7C_2 \times 1 \times ^2C_1 = 42$ | | |
| $3\ \ 1\ \ 1 = ^7C_3 \times ^2C_1 \times ^2C_1 = 140$ | M1 | Summing 3 or 4 options; allow perms, wrong combs but second numbers must sum to 5 etc. |
| Total $= 231$ | A1 **[4]** | Correct answer |

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2 A school club has members from 3 different year-groups: Year 1, Year 2 and Year 3. There are 7 members from Year 1, 2 members from Year 2 and 2 members from Year 3. Five members of the club are selected. Find the number of possible selections that include at least one member from each year-group.

\hfill \mbox{\textit{CAIE S1 2014 Q2 [4]}}
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