How Many Strings Of Four Decimal Digits

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How many strings of four decimal digits

a) do not contain the same digit twice?

b) end with an even digit?

c) have exactly three digits that are 9s?

Explanation

Division rule

If a finite set A is the union of pairwise disjoint subsets with d elements each, then n = A/d.

Product rule

If one event can occur in m ways AND a second event can occur in n ways, the number of ways the two events can occur in the sequence is then m * n.

Subtraction rule

If an event can occur either in m ways OR in n ways (overlapping), the number of ways the event can occur is then m + n decreased by the number of ways that the event can occur commonly to the two different ways.

Sum rule

If an event can occur either in m ways OR in n ways (non-overlapping), the number of ways the event can occur is then m + n.

Solution

A) There are 10 possible digits from 0 to 9.

First digit: 10 ways

Second digit: 9 ways (since the digit cannot be the first digit)

Third digit: 8 ways (since the digit cannot be the first nor second digit)

Fourth digit: 7 ways (since the digit cannot be the first, second nor third digit)

Use the product rule: 10 * 9 * 8 * 7 – 5040

 

B) There are 10 possible digits from 0 to 9.

First digit: 10 ways

Second digit: 10 ways

Third digit: 10 ways

Fourth digit: 5 ways (since there are 5 even digits)

Use the product rule: 10 * 10 * 10 * 5 = 5000

 

C) There are 10 possible digits from 0 to 9.

First nine: 1 way

Second nine: 1 way

Third nine: 1 way

Other digits: 9 ways (since there are 10 digits and the third digit can’t be 9)

Position 4’s: 4 ways (string is 999x, 99×9, 9×99 or x999 with x digit other than 9)

Use the product rule: 1 * 1 * 1 * 9 * 4 = 36

 

Results

A) 5040 strings

B) 5000 strings

C) 36 strings

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