You have just been named manager of the funny car segment of this year\'s civic

Question

You have just been named manager of the funny car segment of this year's civic parade. Your job is to place seven funny cars in parade order. This is more challenging than it looks because you have six antique funny cars and five modern funny cars to choose from (this does mean that four of the cars will not make it into the parade). If the parade chairman decrees that there should be at least two antique and at least two modern funny cars in the parade, how many different parade orders can you make? [Assume that all the funny cars are distinguishable from each other, which is sort of their point!]

Explanation / Answer

The following combinations can be made :

(1) 2 antique cars and 5 modern cars .

(2) 3 antique cars and 4 modern cars .

(3) 4 antique cars and 3 modern cars .

(4) 5 antique cars and 2 modern cars .

Also as all cars are distinct , each combination can be arranged in 7! ways .

Therefore , the total number of parade orders that can be made =

7! . (inom{6}{2} . inom{5}{5} + inom{6}{3} . inom{5}{4} + inom{6}{4} . inom{5}{3} + inom{6}{5} . inom{5}{2})

= 5040 . ( 15 + 100 + 150 + 60)

= 1638000 ways .

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