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 UNORDERED 4entree 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} 
