Answer:
a) 482,718,652,416,000
b) 80,453,108,736,000
Step-by-step explanation:
(a)
There are (combinations of 20 taken 7 at a time)
[tex]\bf \displaystyle\binom{20}{7}=\displaystyle\frac{20!}{7!(20-7)!}=\displaystyle\frac{20!}{7!13!}=77,520[/tex]
different ways of sending 7 jobs to the fastest printer.
There are
13! = 6,227,020,800
ways of sending the other 13 jobs to the rest of the printers (assuming that none of the printers is idle)
By the Fundamental Rule of counting, there are
77520*6,227,020,800 = 482,718,652,416,000
different ways of distributing the jobs with the faster printer getting at least 7 jobs.
(b)
There are (combinations of 20 taken 7 at a time)
[tex]\bf \displaystyle\binom{20}{7}=\displaystyle\frac{20!}{7!(20-7)!}=\displaystyle\frac{20!}{7!13!}=77,520[/tex]
different ways of sending 7 jobs to the fastest printer.
There are (combinations of 13 taken 3 at a time)
[tex]\bf \displaystyle\binom{13}{3}=\displaystyle\frac{13!}{3!(13-3)!}=\displaystyle\frac{13!}{3!10!}=286[/tex]
different ways of sending 3 jobs to the printer that is almost out of paper.
There are
10! = 3,628,800
different ways of sending the 10 remaining jobs to the rest of the printers.
By the Fundamental Rule of counting, there are
77,520*286*3,628,800 = 80,453,108,736,000
different ways to distribute the jobs so that the printer that is almost out of paper does not get more than three jobs and the faster one gets at least 7 jobs.
