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The expression `(20!)/((20-4)!4!)` gives the number of ways 4 items can be picked from a group of 20 items, without replacement, and where order does not matter.
For example if you have a lottery with 20 numbers and you pick 4. You might pick 6,9,11,14. This is the same as picking 9,14,11,6 so order doesn't matter. And once a number is chosen, it cannot be chosen again. (No repetitions, or without replacement).
The symbol ! stands for the factorial operation: 0!=1 by definition, 1!=1,`2! = 2*1,3! = 3*2*1,4! = 4*3*2*1,` etc... It is the product of the number and every natural number below it.
The expression `(n!)/((n-r)!r!)` represents the number of combinations of r items taken from n items without replacement and where order does not matter. This is frequently written as `_nC_r` or `([n],[r])` .
Numerically `(20!)/((20-4)!4!)=4845= ` `_20C_4=(,)` .
** If order matters it is called a permutation and is represented by `_nP_r=(n!)/((n-r)!)` . If you were choosing a committee of 4 people you would use combinations; but if you were selecting officers (president, vice president, etc...) you would use permutations as order matters. -- Alice,Bob,Carol, and Dan can make a committee no matter how you list their names. But if you are selecting officers, it makes a difference whether Alice is president or Carol is president.
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