Combinations and Permutations Practice Problems
Set 1: Factorials
Review the concept of factorials at Combinations and Permutations Help
Practice 1
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.”
Solution 1
The results are shown in Table 8-3. 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 8-3 Values of n ! for n = 0 through n = 15. This table constitutes the solution to Practice 1.
Value of n |
Value of n ! |
0 |
0 |
1 |
1 |
2 |
2 |
3 |
6 |
4 |
24 |
5 |
120 |
6 |
720 |
7 |
5040 |
8 |
40,320 |
9 |
362,880 |
10 |
3,628,800 |
11 |
39,916,800 |
12 |
479,001,600 |
13 |
6,227,020,800 |
14 |
87,178,291,200 |
15 |
1,307,674,368,000 |
Practice 2
Determine the approximate value of 100! using the formula given above.
Solution 2
A calculator is not an option here; it is a requirement. You should use one that has an e ^{x} (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 x ^{y} 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! ≈ (100 ^{100} )/ e ^{100}
≈ (1.00 × 10 ^{200} )/(2.688117 × 10 ^{43} ) ≈ 3.72 10 ^{156}
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.”
Set 2: Permutations and Combinations
Review the concepts of permutations and combinations at Combinations and Permutations Help
Practice 1
How many permutations are there if you have 10 apples, taken 5 at a time in a specific order?
Solution 1
Use the above formula for permutations, plugging in q = 10 and r = 5:
_{10} P _{5} = 10!/(10 – 5)!
= 10!/5! = 10 × 9 × 8 × 7 × 6 = 30,240
Practice 2
How many combinations are there if you have 10 apples, taken 5 at a time in no particular order?
Solution 2
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 _{10} P _{5} = 30,240:
_{10} C _{5} = _{10} P _{5} /5!
= 30,240/120 = 252
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