Question

Question #a91f7

Answer 1

See explanation.

Explanation:

`n` element permutation is a sequenceof `n` elements,

while

`k` element combination of an `n` element set (`n>=k)` is a subset with `k` elements.

The difference between a sequence and a (sub)set is that in a sequence order is important (i.e. `1,2,3` and `3,2,1` are two different sequences), while in the set you only check if the element is there or not, the order does not matter (i.e. `1,2,3` and `3,2,1` are the same sets)

The other difference is that in permutation you only have one number (the number of ordered elements), while combinatin requires 2 numbers (the total number of elements and the number of elements chosen)

I will give some examples to clarify.

Example 1

From a group of `10` boys and `5` girls a team of 2 girls and 2 boys is to be chosen. In how many ways can this be done?

Here we have 2 combinations:

• boys can be choosen in `C""_10^2` ways (a 2 element combination from a 10 element set)

• girls can be chosen in `C""_5^2` ways (a 2 element combination from a 5 element set)

The total number of possible teams is:

`n=C""_10^2xxC""_5^2=(10!)/(2!*8!)xx(5!)/(2!*3!)=`

`=(9*10)/2xx(4*5)/2=45*10=450`

Example 2

In how many ways a set of `4` different books can be put on a shelf?

Here we use permutations because we order the books (i.e. we put them in a sequence)

`n=P_4=4! =24`