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Tuesday, August 27, 2013

Permutation Problems, 3

Category: Algebra, Statistics

"Published in Newark, California, USA"

A passenger train consists of 3 baggage cars, 5 day coaches, and 2 parlor cars. In how many ways can the train be arranged if the 3 baggage cars must come up front?

Solution:

The given word problem is about permutation problem because it involves the number of ways in arranging the objects or things. 

This permutation type is different and it is called a Distinguishable Permutation. 

If a set of n objects consists of k different kinds of objects with n1, objects of the first kind, n2 objects of the second kind, n3 objects of the third kind, and so on, where n1 + n2 + ......... + nk = n, then the number of distinguishable permutations of these objects is 


Now, in the given problem, if

n = 10 train cars in total
n1 = 3 baggage cars
n2 = 5 day coaches
n3 = 2 parlor cars

then, the number of ways in arranging the 10 train cars will be equal to






If the 3 baggage cars must come up front, then the number of ways will be equal to


You have to multiply the previous ways by 3! because the three baggage cars themselves can be arranged in different ways at the front. Therefore, the final answer is