Question

A shop has 5 different printers but there is space for only 3 printers on the display shelf. How many arrangements are possible i. Suppose you have 4 different flags. How many different signals could you make using (i) 2 flags (ii) 2 or 3 flags ii. iii. How many arrangements of the letters of the word B E G I N are there if (i) 3 letters are used (ii) all of the letters are used A quiz team of four is chosen from a group of 15 students. In how many ways could the team be chosen? iv. If there are eight girls and seven boys in a class, in how many ways could a group be chosen so that there are two boys and two girls in the group? v. A school committee consists of six girls and four boys. A social sub-committee consisting of four students is to be formed. In how many ways could the group be chosen if there are to be more girls than boys in the group?

Answer 1

1.

$N=C^3_5\cdot3!=\frac{5!}{3!2!}\cdot3!=60$

2.

i)

$N=2C^2_4=\frac{4!}{2!2!}\cdot2=12$

ii)

$N=2C^2_4+C^3_4\cdot3!=12+\frac{4!}{3!}\cdot3!=36$

3.

i)

$N=C^3_5\cdot3!=\frac{5!}{3!2!}\cdot3!=60$

ii)

$N=5!=120$

4.

$N=C^4_{15}=\frac{15!}{11!4!}=1365$

5.

$N=C^2_7\cdot C^2_8=\frac{7!}{5!2!}\cdot\frac{8!}{6!2!}=21\cdot28=588$

6.

Total number of ways=number of groups with 3 girls + number of groups with 4 girls.

number of groups with 3 girls:

$N_1=C^3_6\cdot C^1_4=\frac{6!}{3!3!}\cdot\frac{4!}{3!}=20\cdot4=80$

number of groups with 4 girls:

$N_2=C^4_6=\frac{6!}{4!2!}=15$

Total number of ways:

$N=N_1+N_2=80+15=95$