We are given four expressions and are asked to identify which of them is different. We can see that every expression is either a permutation or a combination. Let's recall these formulas for n objects taken r at a time!
ccc
Permutations: && Combinations: [0.5em]
_nP_r =n!/( n- r)! && _nC_r =n!/( n- r)!* r!
Now, because the second and third expressions are directly asking for two specific combinations, let's compare their formulas first. We need to show the combinations of 7 objects taken 5 at a time and of 7 objects taken 2 at a time.
Combinations
n!/(n-r)!* r!
_7C_5
7!/2!* 5!
_7C_2
7!/5!* 2!
Because of the Commutative Property of Multiplication, the second and third expressions are equal — their denominators are actually the same. They are also equal to the first expression. By the Transitive Property of Equality, these expressions are all equal to each other. Finally, to see where the fourth expression comes from, let's write the permutations of 7 objects taken 2 at a time.
_7P_2 = 7!/( 7- 2)!
We can see that this equation contains the fourth expression. We can also see that this is different from all previous expressions. Therefore, the fourth expression is the different one.
