Suppose an artists has 9 works of art ready to display, but may only display 4 at a time on a gallery wall.

Question

(a) How many different ways can they arrange 4 paintings side-by-side on the wall?

(b) How many different groups of 4 paintings could they choose to display?

Answer 1

9C4 = 9!/5!4! = 9x8x7x6/4x3x2 = 9x2x7 = 18x7 = 126

Unless order matters

then

9P4 = a whole lot more

= 9!/5! = 9x8x7x6 = 3024

some calculators have combination and permutation keys

where you just plug in 9 and 4 or 5 and out pop the solutions

Answer 2

Problem A asks for the number of permutations as order does matter:

₉P₄ = 9!/(9-4)! = 9!/5! = 9*8*7*6 = 3024 possible ways

Problem B asks for the number of combinations as order doesn't matter:

₉C₄ = 9!/(4!(9-4)!) = 9!/(4!*5!) = (9*8*7*6)/(4*3*2*1) = 3024/24 = 126 groups

Both problems use the following formulas to choose "r" objects from a total of "n" objects:

_nC_r = n!/(r!(n-r)!) = (_nP_r)/r!

Hope this helped!