COMBINATiONS AND PERMUTATIONS

Combinations refer to the selection of items from a larger set where the order does not matter. For example, choosing 2 fruits from a basket of 5 different fruits.

Permutations refer to the arrangement of items from a larger set where the order does matter. For example, arranging 3 books on a shelf from a selection of 5 books.

The formula for combinations is given by: $$C(n, r) = \frac{n!}{r!(n-r)!}$$ where n is the total number of items, and r is the number of items to choose.

The formula for permutations is given by: $$P(n, r) = \frac{n!}{(n-r)!}$$ where n is the total number of items, and r is the number of items to arrange.

There are $$36^6$$ unique license plates possible, since there are 36 options (10 digits + 26 letters) for each of the 6 positions.

There are $$10^3 = 1,000$$ different 3-digit passwords possible.

Mark can wear $$5 \times 7 = 35$$ different pant/shirt combinations.

The number of distinguishable arrangements is calculated using the formula: $$\frac{n!}{n_1! \times n_2! \times ...}$$ where n is the total letters and n_1, n_2 are the frequencies of each letter.

The total number of arrangements is $$\frac{11!}{2! \times 2! \times 2!} = 4,986,000$$ .

The key difference is that in combinations, the order of selection does not matter, while in permutations, the order of arrangement does matter.