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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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For each exercises 37-39, the number of elements in a certain set can be found by computing the number in a larger universe that are not in the set and subtracting this from the total in the larger universe. In each of these, as was the case the solution to Example 9.3.6(b). De Morgan's laws and the inclusion/exclusion rule can be used.
39. How many integers from 1 through 999,999 contain each of the digits 1, 2, and 3 at least once? (Hint: For each of the digits 1, 2, and 3 at least once? all integers from 1 through 999,999 that do not contain the digit i.)

To determine

To find the number of integers from 1 through 999,999 contain each of the digits 1,2and 3  at least once.

Explanation

Given information:

DeMorgan’s laws and inclusion /exclusion rule can be used.

Concept used:

n(AUB)=n(A)+n(B)n(AB)

Calculation:

Let us find out the number of integers from 1 through 999,999 contain each or the digits 1, 2 and 3 at least once.

Now, let us find out the total integers which do not contain either of 1, 2 or 3.

Case 1: Let us find out total number of integers which do not contain digit 1.

Therefore, the available digits are from 0 to 9 except 1

Six digits number can be formed using these 9 digits with repetition is as follows

961 (Exclude 0 from the total numbers)

Similarly, total number of integers which do not contain digit 2or3 is also 961

Case 2: Let us find out total number or integers which do not contain digit 1 and 2.

Therefore, the available digits are from 0 to 9 except 1 and 2

Six digits number can be formed using these 8 digits with repetition is as follows

861 (Exclude 0 from the total numbers)

Similarly, total number of integers which do not contain digit 2 and 3 or 1 and 3 is also 861.

Case 3: Let us find out total number of integers which do not contain digit 1, 2 and 3

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