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A palindromic number is a number that remains the same when its digits

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New post 10 Jan 2018, 01:10
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[GMAT math practice question]

A palindromic number is a number that remains the same when its digits are reversed. For example, \(16461\) is a palindromic number. If a \(4\) digit integer is selected randomly from the set of all \(4\) digit integers, what is the probability that it is palindromic?

A. \(\frac{1}{20}\)
B. \(\frac{1}{50}\)
C. \(\frac{1}{60}\)
D. \(\frac{1}{90}\)
E. \(\frac{1}{100}\)

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Re: A palindromic number is a number that remains the same when its digits  [#permalink]

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New post 10 Jan 2018, 02:15
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Total 4-digit integers= 9999-1000+1= 9000.
There is only one palindromic number in evey 100 nos.
Therefore, total no.of palindromic= 90.
Probability= 90/9000=1/100.

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Re: A palindromic number is a number that remains the same when its digits  [#permalink]

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New post 16 Jan 2018, 17:48
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MathRevolution wrote:
[GMAT math practice question]

A palindromic number is a number that remains the same when its digits are reversed. For example, \(16461\) is a palindromic number. If a \(4\) digit integer is selected randomly from the set of all \(4\) digit integers, what is the probability that it is palindromic?

A. \(\frac{1}{20}\)
B. \(\frac{1}{50}\)
C. \(\frac{1}{60}\)
D. \(\frac{1}{90}\)
E. \(\frac{1}{100}\)


Let’s determine the number of the 4-digit palindromes. Notice that a 4-digit number is a palindrome if it’s one of the following two formats: XXXX and XYYX where X and Y represent a digit and X ≠ Y and X is nonzero.

Format 1: XXXX

We see that X can be any digit from 1 to 9, inclusive; thus, there are 9 such numbers.

Format 2: XYYX

We see that X can be any digit from 1 to 9, inclusive, and Y can be any digit from 0 to 9, inclusive (excluding digit X), so that there are 9 choices for X and 9 choices for Y; thus, the number of 4-digit number in this format is 9 x 9 = 81.

Thus, there are a total of 9 + 91 = 90 numbers in both formats. Since there are 9000 four-digit numbers (1000 to 9999 inclusive), the probability of picking a 4-digit palindrome randomly is 90/9000 = 1/100.

Answer: E
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Re: A palindromic number is a number that remains the same when its digits  [#permalink]

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New post 10 Jan 2018, 02:28
You can choose first and second digit randomly (first digit not zero). Chance that third digit equals second: 1/10. Equally, chance that fourth digit = first digit: 1/10. That 0 is not possible for first digit doesn't matter: once chosen, chance that fourth is the same = 1/10. Check: other way round: choose 3rd and 4th at random. Chance that 1st equals 4th: 0 if 4th=0 and 1/9 if 4th not 0. So: chance that first equals 4th: 1/10 *0 + 9/10 * 1/9 = 1/10. So answer: 1/10 * 1/10 = 1/100

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New post 12 Jan 2018, 00:52
=>

4-digit palindromic numbers have the form \(‘xyyx’\), where
\(x\) is one of values \(1,2,…,9\) and \(y\) is one of values \(0,1,2,…,9.\)

So, there are \(9 * 10 = 90\) four-digit palindromic numbers.
The total number of 4-digit numbers between \(1000\) and \(9999\), inclusive, is \(9000 ( = 9999 – 1000 + 1 )\).

Therefore, the probability that the selected 4-digit is palindromic is \(\frac{90}{9000} = \frac{1}{100}.\)

Therefore, the answer is E.
Answer: E
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Re: A palindromic number is a number that remains the same when its digits  [#permalink]

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Re: A palindromic number is a number that remains the same when its digits   [#permalink] 24 Feb 2019, 21:07
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