Permutations and Combinations Help (page 2)
In probability, it is often necessary to choose small sets from large ones, or to figure out the number of ways in which certain sets of outcomes can take place. Permutations and combinations are the two most common ways this is done. Before we can define these, however, we need to define a function of non-negative integers called the factorial.
The factorial of a number is indicated by writing an exclamation point after it. If n is a natural number and n ≥ 1, the value of n! is defined as the product of all natural numbers less than or equal to n:
- n! = 1 × 2 × 3 × 4 × . . . × n
If n = 0, then by default, n! = 1. The factorial is not defined for negative numbers.
As n increases, the value of n! goes up rapidly, and when n reaches significant values, the factorial skyrockets. There is a formula for approximating n! when n is large:
- n! ≈ nn / en
where e is a constant called the natural logarithm base, and is equal to approximately 2.71828. The squiggly equals sign emphasizes the fact that the value of n! using this formula is approximate, not exact.
Permutations and Combinations
Suppose q and r are both positive integers. Imagine a set of q items taken r at a time in a specific order. The possible number of permutations in this situation is symbolized qPr and can be calculated as follows:
- qPr = q! / (q – r)!
Suppose q and r are positive integers. Imagine a set of q items taken r at a time in no particular order. The possible number of combinations in this situation is symbolized qCr and can be calculated as follows:
- qCr = qPr / r! = q! / [r!(q – r)!]
Permutations and Combinations Practice Problems
Write down the values of the factorial function for n = 0 through n = 15, in order to illustrate just how fast this value ''blows up.''
The results are shown in Table 3-3. It's perfectly all right to use a calculator here. It should be capable of displaying a lot of digits. Most personal computers have calculators that are good enough for this purpose.
Table 3-3 Values of n! for n = 0 through n = 15. This table constitutes the solution to Practice 1.
Determine the approximate value of 100! using the formula given above.
A calculator is not an option here; it is a requirement. You should use one that has an ex (or natural exponential) function key. In case your calculator does not have this key, the value of the exponential function can be found by using the natural logarithm key and the inverse function key together. It will also be necessary for the calculator to have an xy key (also called x^y) that lets you find the value of a number raised to its own power. In addition, the calculator should be capable of displaying numbers in scientific notation, also called power-of-10 notation. Most personal computer calculators are adequate if they are set for scientific mode.
Using the above formula for n = 100:
- 100! ≈ (100100) / e100
- ≈ (1.00 × 10200) / (2.688117 × 1043)
- ≈ 3.72 × 10156
The numeral representing this number, if written out in full, would be a string of digits too long to fit on most text pages without taking up two or more lines. Your calculator will probably display it as something like 3.72e + 156 or 3.72 E 156. In these displays, the ''e'' or ''E'' does not refer to the natural logarithm base. Instead, it means ''times 10 raised to the power of.''
How many permutations are there if you have 10 apples, taken 5 at a time in a specific order?
Use the above formula for permutations, plugging in q = 10 and r = 5:
- 10P5 = 10! / (10 – 5)!
- = 10! / 5!
- = 10 × 9 × 8 × 7 × 6
- = 30,240
How many combinations are there if you have 10 apples, taken 5 at a time in no particular order?
Use the above formula for combinations, plugging in q = 10 and r = 5. We can use the formula that derives combinations based on permutations, because we already know from the previous problem that 10P5 = 30,240:
- 10C5 = 10P5 / 5!
- = 30,240 / 120
- = 252
Practice problems for these concepts can be found at:
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