A Billion

Probability Level 3

How many ten-digit numbers have at least two equal digits?


The answer is 8996734080.

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Arpit MIshra by Arpit MIshra , 21, India

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3 solutions

Arpit MIshra
Apr 7, 2015
  • Number of ten-digit numbers = 9 × 1 0 9 9 \times 10^{9}
  • Number of ten-digit numbers with no equal digits = 9 × 9 ! 9 \times 9!

Therefore, Number of ten-digit numbers with at least two equal digits = 9 × 1 0 9 − 9 × 9 ! 9 \times 10^{9} - 9 \times 9!

= 8996734080 \boxed{8996734080}

12 Helpful 0 Interesting 0 Brilliant 0 Confused
Paola Ramírez
Apr 18, 2015

Numbers of ten-digiit numbers- Number of ten-digit numbers whose digits do not repeat = 9 × 1 0 9 − 9 ( 9 ! ) = 8996734080 9\times 10^9-9(9!)=\boxed{8996734080}

Inclusion-exclusion

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9 ∗ 1 0 9 − 10 ! + 9 ! = 8996734080 9*10^9-10!+9!=\boxed{8996734080}

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