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

[BTGModeratorVI]

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

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

Answer: A
Source: Math Revolution

[Jay@ManhattanReview]

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

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

Answer: A
Source: Math Revolution

Since we need to find the numbers that have at least one 0 digit, we can get this calculated in a smarter way by first calculating the numbers that have no 0 digit and then deducting their count from the total no. of 5-digit numbers.

Let's first calculate the total number of 5-digits = 9*10*10*10*10 = 90,000; I took the no. of ways of filling the ten-thousands digit equal to 9 since no 5-digit number can start from 0.

No. of 5-digit numbers that have no 0 = 9*9*9*9*9 = 9^5 = 59,049

The required no. = 90,000 – 59,049 = 30,951

The correct answer: A

Hope this helps!

-Jay
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[Brent@GMATPrepNow]

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

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

Answer: A
Source: Math Revolution

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 SonalSinha803 shows us above, the correct answer is 90,000 - 9^5

However, 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