Question

# 1-How many different ways to make a fruit plate are there, if there are 5 apples,...

1-How many different ways to make a fruit plate are there, if there are 5 apples, 4oranges, and 3 pears, and the plate has to contain at least 3 pieces of fruit?

2 -Pilot has to be on 7 round trip flights in April, one round trip flight per day.How many ways are there to schedule his flights so that no two flights occur ontwo consecutive dates?

1)

Number of ways of choosing at least 3 apples out of 5 = \binom{5}{3} + \binom{5}{4} + \binom{5}{5} = 10 + 5 + 1 = 16

Number of ways of choosing at least 3 out of 4 oranges = \binom{4}{3} + \binom{4}{4} = 4 + 1 = 5

Number of ways of choosing at least 3 pears out of 3 = 1

Thereofre, total number of ways to do the task of making a fruit plate such that htere are 3 pieces of each fruit in it

= 16*5 *1 = 80

2) The number of ways to do so

= Day 1 has 1 out of 7 roud trips, day 2 has none of the 6 trips, day 3 has 1 of the 6 trips, day 4 has none of the 5 trips. ... day 17 has 1 out of the remaining 1 trip

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