Question

How can you determine if you need to use a combination or permutation to count the number of outcomes? Which will usually have more outcomes? Why? Provide an example in your explanation.

Answer 1

Answer with Step-by-step explanation:

Permutation : It is an arrangement of r elements out of n elements.

Combination : it is a selection of r element out of n elements .

Suppose we have a set

S={1,2,3}

If two elements are taken at a time then

Using permutation formula

Total number of outcomes=$3P_2$

Total number of outcomes=$\frac{3!}{(3-2)!}$

Total number of outcomes=3!=$3\times 2\times1=6$

Using combination formula

$\binom{n}{r}=\frac{n!}{r!(n-r)!}$

Total number of outcomes=$\binom{3}{2}=\frac{3!}{2!1!}$

Total number of outcomes=$\frac{3\times2!}{2!}$

Hence, total number of outcomes=3

Total number of outcomes determined by permutation have more outcomes.

Because permutation is an arrangement of elements  therefore, it consider order of arrangement of element   but combination is a selection of elements it does no consider order of elements

Arrangements of two elements out of 3 elements

{1,2},{2,3},{2,1},{3,2},{1,3},{3,1}

By using combination if two elements taken at a time then combination

{1,2},{2,3},{1,3}