Question 145215
a){{{6*5*4*3*2*1 = 720}}}
b) You're looking for combinations here, not arrangements, because
you don't care how the prize winners are arranged, once they have
their prizes. You need the formula for combinations of 3 things
chosen from 6
{{{6! / (3! *(6-3)!) = 20}}}
It's easy to check:
Call the entries ABCDEF
The possible combinations:
Including A
ABC ACD ADE AEF
ABD ACE ADF
ABE ACF
ABF
Including B (without repeats)
BCD BDE BEF
BCE BDF
BCF
Including C (without repeats)
CDE CEF
CDF
Including D (without repeats)
DEF
(Note that E and F are already combined with A,B,C,and D) 
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These add up to {{{20}}} combinations