Alber theory!

Probability Level 4

The number of positive three digit numbers that have exactly two consecutive digits which are equal, is n n . Find the sum of all natural numbers till n n .


The answer is 13203.

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Prateek Mehra by Prateek Mehra , 22, India

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2 solutions

Prateek Mehra
Jan 10, 2015

If we take x as the hundreds digit- Number of numbers of the form x00 = 9, number of numbers of the form x11 = 8(we can't use 0 and 1) Thus we get 17 for x11 and we will get 17 for each of x22, x33,x44,x55, x66, x77, x88,x99 Hence n= 9 + 17*9 = 162. Then if we use Arithmetic progression to find out the sum of all natural numbers till n,we will get 13203 as our answer.

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I have rephrased your question for clarity. Can you check that the original intention is still intact?

Calvin Lin Staff - 6 years, 5 months ago

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Yes it is.Thanks

Prateek Mehra - 6 years, 5 months ago
Christian Barrera
Jan 29, 2015

Let ABC represent the 3 digit numbers.

If the first 2 digits are the same, A=B, it becomes AAC:

Count of AAC numbers: 9 possible options for AA (any number from 1-9), 9 possible options for C (any number from 0-9 except the number used in A)

= 9 * 9 = 81

If the last 2 digist are the same, B = C, it becomes ABB:

Count of ABB numbers: 9 possible options for A (any number from 1 to 9), 9 possible options form B (any number from 0-9 except the number used in A)

= 9 * 9 = 81

Total Count of Possible Numbers = 162

Sum from 1 to 162 = (1+162)*162/2 = 13203

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