Solution to Chinese Restaurant Problem

THE CHINESE RESTAURANT

re-stated problem
A Chinese restaurant charges a standard price for it's "family dinner", consisting of any four different dishes (or entrees) on it's menu. The restaurant's advertisement claims that "over 3OO different family dinners are possible". If this claim is true, what must be the smallest possible number of dishes (entrees) listed on the menu?

re-stated problem
A Chinese restaurant charges a standard price for it's "family dinner", consisting of any four different dishes (or entrees) on it's menu. The restaurant's advertisement claims that "over 3OO different family dinners are possible". If this claim is true, what must be the smallest possible number of dishes (entrees) listed on the menu?

Solution: Let n=number of entrees on the restaurant menu

Since a "family dinner" is an UN-ORDERED 4-entree subset
of the set of n entrees on the menu, therefore:

C(n,4)

number of possible dinners

Using the above factorial formula for various values of n, we get:

C(4,4)	=	1
C(5,4)	=	5
C(6,4)	=	15
C(7,4)	=	35
C(8,4)	=	70
C(9,4)	=	126
C(10,4)	=	210
C(11,4)	=	330

Thus, the minimum value of n is 11.

One calculation is shown here: _C(10,4) ₌     10!     ₌ 10x9x8x7 x (6!)
4! x 6! 4x3x2 x (6!)

₌    10x9x8x7   ₌   10x9x7   ₌ ₂₁₀
^4x3x2 ³


This page last updated
20 June 1998

Solution:	Let n=number of entrees on the restaurant menu

	Since a "family dinner" is an UN-ORDERED 4-entree subset of the set of n entrees on the menu, therefore:

One calculation is shown here:	_C(10,4)	₌	10!	₌	10x9x8x7 x (6!)
			4! x 6!		4x3x2 x (6!)

	₌ 10x9x8x7	₌	10x9x7	₌ ₂₁₀
	^4x3x2		³