A2 October 2020 Q1

EdexcelCurrent spec6 marksNumber Theory

1. A small sports club has 12 adult members and 14 junior members.

The club needs to enter a team of 8 players for a particular competition.

Determine the number of ways in which the team can be selected if

(i) there are no restrictions on the team, (1)

(ii) the team must contain 4 adults and 4 juniors, (2)

(iii) more than half the team must be adults. (3)

Scheme	Marks	AO
\({}^{26}C_{8} = 1562275\)	B1	1.1b
	(1)

Scheme	Marks	AO
Attempts \({}^{12}C_{4} \times {}^{14}C_{4}\)	M1	1.1b
\(= 495 \times 1001 = 495495\)	A1	1.1b
	(2)

Mark scheme (iii)

Scheme	Marks	AO
Attempts cases for 5, 6, 7 or 8 adults on team (only).	M1	3.1a
\({}^{12}C_{5} \times {}^{14}C_{3} + {}^{12}C_{6} \times {}^{14}C_{2} + {}^{12}C_{7} \times {}^{14}C_{1} + {}^{12}C_{8} \times {}^{14}C_{0} = \ldots\)	M1	1.1b
= 383955	A1	1.1b
	(3)
	(6 marks)

Notes

M1: Works out the different cases that give more than half adults – e.g. attempts to find combinations for 5, 6, 7 or 8 adults (or 0, 1, 2 or 3 juniors). Need not be the correct formula, look for evidence of the correct cases considered. Do not allow this mark if the case of 4 adults and 4 juniors is included.

M1: Sums the possibilities. Allow if the 4 adults/4 juniors case is included, but the binomial products must be correct but e.g. \({}^{14}C_{0}\) may not be seen (as it is 1). Must be an attempt at using the correct formulae.

A1: cao 383955