How many 5 digit numbers have at least o

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[GMAT math practice question]

How many 5 digit numbers have at least one zero digit?

A. 30951
B. 40141
C. 47132
D. 50001
E. 50433

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Max@Math Revolution wrote:[GMAT math practice question]

How many 5 digit numbers have at least one zero digit?

A. 30951
B. 40141
C. 47132
D. 50001
E. 50433
One way to answer the question is to find the total number of five-digit numbers, and then subtract the number of five-digit numbers that have NO zero-digits. The difference is the number of five-digit numbers having AT LEAST ONE zero-digit.

Find the total number of five-digit numbers:
The first digit can be any number from 1 to 9 (if the first digit were 0, it would be a four-digit number), and each of the next four digits can be any number from 0 to 9.
Therefore, there are 9 possible values of the first digit and 10 possible values for each of the remaining digits.
The total number of five-digit numbers is 9 × 10 × 10 × 10 × 10 = 90,000.

Find the number of five-digit numbers that have NO zero-digits:
Each of the 5 digits can be any number from 1 to 9.
Therefore, the number of five-digit numbers that include NO zero-digits is 9 × 9 × 9 × 9 × 9 = 59,049.

Find the difference:
90,000 - 59,049 = 30,951.

The correct answer is choice A

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by Brent@GMATPrepNow » Mon Apr 23, 2018 10:02 am
Max@Math Revolution wrote: How many 5 digit numbers have at least one zero digit?
A. 30951
B. 40141
C. 47132
D. 50001
E. 50433
NOTE: We can solve this question without performing lengthy calculations (e.g., 9^5)

# of 5-digit numbers with at least one zero digit = (TOTAL number of 5-digit numbers) - (number of 5-digit numbers without ANY 0's)
As Keith shows us above, we get:
# of 5-digit numbers with at least one zero digit = 90,000 - 9^5

Before we calculate 9^5 (and then subtract that value from 90,000), we should keep in mind that the GMAT test-makers are NOT interested in our ability to perform lengthy/tedious calculations (for more on this, read https://www.gmatprepnow.com/articles/re ... st-maker-0)

We already know that the answer to the question is 90,000 - 9^5
Notice that this value is a multiple of 9, since 90,000 - 9^5 = 9(10,000 - 9^4)
So, let's check the answer choices so see which one (or ones) is divisible by 9

RULE: If a number is divisible by 9, then the sum of its integers is divisible by 9.

A. 30951 Sum of digits = 18, which is divisible by 9. Since, answer choice A is divisible by 9, we'll KEEP IT.
B. 40141 Sum of digits = 10, which is NOT divisible by 9. Since, answer choice B is NOT divisible by 9, we'll ELIMINATE IT.
C. 47132 Sum of digits = 17, which is NOT divisible by 9. Since, answer choice C is NOT divisible by 9, we'll ELIMINATE IT.
D. 50001 Sum of digits = 6, which is NOT divisible by 9. Since, answer choice D is NOT divisible by 9, we'll ELIMINATE IT.
E. 50433 Sum of digits = 15, which is NOT divisible by 9. Since, answer choice E is NOT divisible by 9, we'll ELIMINATE IT.

By the process of elimination, the correct answer is A

Cheers,
Brent
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by Scott@TargetTestPrep » Tue Apr 24, 2018 10:45 am
Max@Math Revolution wrote:[GMAT math practice question]

How many 5 digit numbers have at least one zero digit?

A. 30951
B. 40141
C. 47132
D. 50001
E. 50433
To solve, let's do the "opposite." That is, let's determine the number of 5-digit numbers that have no 0 digits.

If none of the 5 digits is 0, then we have 9 x 9 x 9 x 9 x 9 = 9^5 = 59,049 five-digit numbers without a 0 in it. Since there are 90,000 five-digit numbers (10,000 to 99,999), we have

90,000 - 59,049 = 30,951

integers that have at least one digit as 0.

Answer: A

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by Max@Math Revolution » Tue Apr 24, 2018 11:26 pm
=>

The number of 5 digit numbers is 99999 - 10000 + 1 = 90000.
The number of 5 digit numbers which have no 0 digit is 9 * 9 * 9 * 9 * 9 = 59049.
When we encounter the words "at least" in probability questions, we should consider the complementary case and subtract the number of ways it occurs from the total number of possibilities.
The total number of 5 digit numbers with at least one zero digit is
90000 - 59049 = 30951

Therefore, A is the answer.
Answer A