This can be done using the choose function.

The number of combinations are given by:

#([n],[k])=(n!)/(k!(n-k)!)#

where #n# is the total number of students and #k# is the number of students to be picked. So we have #n=20# and #k=5#:

#([20],[5])=(20!)/(5!(20-5)!)=(20!)/(5!15!)#

Evaluate directly with a calculator:

#(20!)/(5!15!)=15504#

we can simplify this before calculation by hand:

#(20!)/(5!15!)=(20times19times...times2times1)/(5times4times3times2times1times(15times...times1)#

#=((20times...times16)(15times...times1))/((5times...times1)(15times...times1))=((20times...times16)cancel(15times...times1))/((5times...times1)cancel(15times...times1))#

#=((color(red)20times19timescolor(blue)18times17timescolor(green)16))/((color(red)5timescolor(green)4times3timescolor(blue)2times1))#

Simplify the numbers matched up by color:

#=((4times19timescolor(green)9times17times4))/((1times1timescolor(green)3times1times1))#

#=((4times19times3times17times4))/((1times1times1times1times1))#

#=3times4times4times17times19#

#=48times323=15504#