- Combinations

A combination is a selection of objects in which the order of selection does not matter. The number of different combinations of \(r\) distinct objects out of \(n\) distinct objects is

\[^nC_r ={n!\over (n-r)!r!}\]
this result only holds for objects that are different.
LESSON 1
In how many ways can three letters be chosen from A, B, C and D?
SOLUTION
\(^4C_3 =4\)

LESSON 2
Calculate the number of ways of choosing 5 letters from the letters of the word PRINGLES.
SOLUTION
\(^8C_5 =56\)

LESSON 3
A group of 9 people consists of 5 boys and 4 girls. In how many ways can a team of 3 boys and 3 girls be selected.
SOLUTION
\(^5C_3\times ^4C_3 =40\)

LESSON 4
Find the number of ways of choosing a school team of 5 pupils from 6 boys and 8 girls
(i) if there are more girls than boys in the team,
(ii) if three of the boys are cousins and are either all in the team or all not in the team.
SOLUTION
(i) There are 3 considerations
0 boys and 5 girls – \(^6C_0 \times ^8C_5=56\)
1 boy and 4 girls – \(^6C_1\times ^8C_5=6×70=420\)
2 boys and 3 girls – \(^6C_2×^8C_3 =15×56=840\)
Number of ways is \(56+420+840=1,316\)
(ii) ALL IN
3 cousins and 2 from the remaining 11
\(3×^{11}C_2=55\)
ALL OUT
0 of the cousins and 5 from the remaining 11
\(^{11}C_5=462\)
Total combinations: \(55+462=517\)

LESSON 5
A committee of 7 people is to be chosen at random from 18 volunteers.
(i) In how many ways can the committee be chosen?
The 18 volunteers consist of 5 people from Barbados, 6 from Trinidad and 7 from Guyana. The committee is to be chosen randomly. Find the number of committees which will
(ii) consist of 2 people from Barbados, 2 people from Trinidad and 3 people from Guyana,
(iii) include exactly 5 people from Guyana,
(iv) include at least 2 people from each of the three islands.
SOLUTION
(i) \(^{18}C_7=31,824\)
(ii) \(^5C_2× ^6C_2×^7C_3=5,250\)
(iii) \(^7C_5×^{11}C_2=1,155\)
(iv) 2 from Barbados, 2 from Trinidad, 3 from Guyana
\(^5C_2×^6C_2×^7C_3=5,250\)
2 from Barbados, 3 from Trinidad, 2 from Guyana
\(^5C_2 ×^6C_3×^7C_2=4,200\)
3 from Barbados, 2 from Trinidad, 2 from Guyana
\(^5C_3×^6C_2×^7C_2=3,150\)
Total committees is 12,600