| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Topic | Combinations & Selection |
| Type | Arranging identical items in a line |
| Difficulty | Standard +0.3 Part (i) is a standard permutations with repetition problem (7!/2! = 2520). Part (ii) requires casework based on whether the two 1's are both selected, adding modest complexity but still following textbook methods. This is slightly above average difficulty due to the casework in part (ii), but remains a routine combinatorics question. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
**Question 5**
**(i)**
$\frac{7!}{2!} = \frac{5040}{2} = 2520$ — M1, M1, A1 **[3]**
$7!\ \div 2!$; c.a.o.
**(ii)**
$^6C_5$ — M1 (Consider selections when all digits are different)
$^6C_5 \times\ ^5P_5$ or $^6P_5 = 720$ — A1 (Arrangements when all digits different)
$^5C_3$ — M1 (Consider selections of the form 11xxx)
$(10) \times \frac{5!}{2!} = 600$ — M1, A1 (Arrangements of 11xxx)
$720 + 600$ — M1 (Adding two (or more) relevant cases)
$= 1320$ — A1 **[7]**
OR: (e.g.) Using no 1's + one 1 + two 1's
$= \ ^5P_5 + 5 \times\ ^5P_4 + 10 \times\ ^5P_3$
$= 120 + 600 + 600 = 1320$
**Total: [10]**5 A game is played with cards, each of which has a single digit printed on it. Eleanor has 7 cards with the digits $1,1,2,3,4,5,6$ on them.\\
(i) How many different 7-digit numbers can be made by arranging Eleanor's cards?\\
(ii) Eleanor is going to select 5 of the 7 cards and use them to form a 5 -digit number. How many different 5-digit numbers are possible?
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2013 Q5 [10]}}