Solutions — Counting 10C Worksheet

Year 11 Mathematical Methods · Cambridge Methods 1&2, Counting Methods · Mr Wong

ANSWER KEY

Warm-up — selection or arrangement?

Q1. Type then value:

a S — $^{52}C_5$

b A — $5!=120$

c S — $^8C_3=56$

d A — $4!=24$

Q2. Evaluations:

a $^8C_3=56$

b $^{10}C_4=210$

c $^{13}C_7=1716$

Part A — Basic selections

3a $^{13}C_7=1716$ teams

3b Choosing $7$ to keep is the same as choosing the $6$ to leave out, so $^{13}C_7=\,^{13}C_6$ (both $=1716$).

4 $^{20}C_3=1140$ sundaes

5 $^{15}C_{11}=1365$ teams ($=\,^{15}C_4$)

Part B — Some guaranteed & at least

6 $^9C_3=84$ committees (fix the named person, choose 3 more from 9)

7a $^5C_2\times\,^5C_3=10\times10=100$ (exactly 2 boys)

7b $^{10}C_5-\,^5C_5=252-1=251$ (at least 1 boy)

8 $^4C_2{}^6C_2+{}^4C_3{}^6C_1+{}^4C_4{}^6C_0=90+24+1=115$ (2, 3, 4 women)

Part C — Selection of any size

9a $2^6=64$ selections (incl. none)

9b $2^6-1=63$ (at least one)

Challenge

10a $^5C_2\times\,^7C_3\times\,^6C_2=10\times35\times15=5250$ delegations

10b The delegation is a junior group and a middle group and a senior group — three independent choices made together — so by the multiplication principle the counts multiply (adding would count only one group at a time).