11.3 Each of the 7 children in turn, throws a ball once at a target. Calculate the number of ways the children can be arranged in order to take the throws.
Given that 3 of the children are girls and 4 are boys, calculate the number of
ways the children can be arranged in order that
a) successive throws are made by boys and girls alternatively
b) a girl takes the first throw and a boy takes the last throw.
a) To have successive throws made by boys and girls alternatively, we can arrange the 4 boys and 3 girls in a pattern such that the first child is a boy. There are 4 boys and 3 girls who can take the first throw, so there are 4 choices for the first position. After the first child, the pattern alternates between boys and girls, so there are 3 choices for the second position (girl), 3 choices for the third position (boy), and so on. Therefore, the total number of arrangements is 4 * 3 * 3 * 2 * 2 * 1 * 1 = 144 ways.
b) If a girl takes the first throw and a boy takes the last throw, we can treat these two positions as fixed. Therefore, we need to arrange the remaining 2 boys and 3 girls in the remaining 5 positions. There are 2 choices for the second position (boy), 3 choices for the third position (girl), 2 choices for the fourth position (boy), and 2 choices for the fifth position (girl). Therefore, the total number of arrangements is 2 * 3 * 2 * 2 * 1 = 24 ways.