Question

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?

Answer 1

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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