Solution to Problem 116


Congratulations to this week's winner

Kenny Albright

No other correct solutions were received, though a number of partial solutions were offered in non-closed form.  Further correct solutions came from Rick Bischoff, Ken Duisenberg, Yves Thiry, Francesc Suñol i Esquirol, Irwin Tolins, Ahron Teitelman, Sudipta Das, Philippe Fondanaiche.

The exact number of guests is the integer part of the number 100e - 1, where e = 2.718... is Euler's number. 

It's easy to see that the total number of guests is given by the sum

100 + (100)(99) + (100)(99)(98) + ... + (100)(99)(98)...(2)(1)
= 

The approximation of the summation, S, by Euler's number is based on the Taylor series expansion of the function ex.  (See your favorite text on Calculus for further information on this.) This approximation of e is very accurate, but the error in the approximation is being multiplied by 100! which is pretty big.  We can see just how big this error is by using the Lagrange form of the error, and a calculator for some integration; namely

(You might have to look in a couple of Calculus texts for this.)  You conclude that the total number of guests is the integer part of 100e - 1.

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Page last updated 15 May 2001.

ã 2001 Alberto L Delgado