Off-peak Bus Journey

Level pending

6 (indistinguishable) people get on to an empty bus as it begins its route, and no one else boards the bus once it has set off. There are 10 (distinguishable) stops where the passengers may get off the bus. When the bus reaches the end of the line it must be empty. Let n be the number of ways that the passengers can leave the bus. What is the sum of the digits of n ?

Details and Assumptions \textbf{ Details and Assumptions }

It is the number of people that get off at each of the 10 stops which matters, not who the people are who get off


The answer is 10.

Josh Rowley by Josh Rowley , 24, United Kingdom

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1 solution

Anish Puthuraya
Feb 5, 2014

Let,
Number of people who get off at Stop 1 = x 1 = x_1
Number of people who get off at Stop 2 = x 2 = x_2
⋮ \vdots
Number of people who get off at Stop 10 = x 10 = x_{10}


It is clear that,
x 1 + x 2 + … + x 10 = 6 x_1+x_2+\ldots+x_{10} = 6
where, x 1 , x 2 , … , x 10 ≥ 0 x_1,x_2,\ldots,x_{10}\geq 0

Thus, the number of integral solutions to this equation are,
( 6 + 10 − 1 10 − 1 ) {6+10-1}\choose{10-1} = = ( 15 9 ) {15}\choose{9} = = 5005 5005

Hence, the digit sum of 5005 5005 is,
5 + 0 + 0 + 5 = 10 5+0+0+5 = \boxed{10}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Hey can u please tell me..How did u find out the no of integral solutions

ajinkya Kulkarni - 7 years, 2 months ago

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