Question

From a group of 8 boys and 6 girls, 6 children are to be selected. In how many different ways can they be
selected so that at least one boy should be there?​

Answer 1

Answer:

3814 ways

Step-by-step explanation:

Number of boys = 8

Number of girls = 6

Number of children to be selected = 6

Condition: At least 1 boy in the selected group.

Ways to select 6 children with 1 boy:

$8C_{1}$ x $6C_{5}$

Ways to select 6 children with 2 boys:

$8C_{2}$ x $6C_{4}$

Ways to select 6 children with 3 boys:

$8C_{3}$ x $6C_{3}$

Ways to select 6 children with 4 boys:

$8C_{4}$ x $6C_{2}$

Ways to select 6 children with 5 boys:

$8C_{5}$ x $6C_{1}$

Ways to select 6 children with 6 boys:

$8C_{6}$ x $6C_{0}$

Total number of ways = $8C_{1}$ x $6C_{5}$ + $8C_{2}$ x $6C_{4}$ + $8C_{3}$ x $6C_{3}$ + $8C_{4}$ x $6C_{2}$ + $8C_{5}$ x $6C_{1}$ + $8C_{6}$ x $6C_{0}$

= 8 x [(6 x 5 x 4 x 3 x 2 x 1)/(5 x 4 x 3 x 2 x 1)] + [(8 x 7)/(2 x 1)] x [(6 x 5 x 4 x 3)/(4 x 3 x 2 x 1)] +  [(8 x 7 x 6)/(3 x 2 x 1)] x [(6 x 5 x 4)/(3 x 2 x 1)] +  [(8 x 7 x 6 x 5)/(4 x 3 x 2 x 1)] x [(6 x 5)/(2 x 1)] +  [(8 x 7 x 6 x 5 x 4)/(5 x 4 x 3 x 2 x 1)] x [(6)/(1)] +  [(8 x 7 x 6 x 5 x 4 x 3)/(6 x 5 x 4 x 3 x 2 x 1)]

= 48 + 420 + 1120 + 1050 + 336 + 840

= 3814 ways