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10 votes
10 votes

 

NOTE- Here digits can repeat, it was the mistake i did during test :)

6 votes
6 votes

There are $2, 250.$ The reason is as follows. Let $\text{A}$ be the set of four-digit numbers that are odd. Let $\text{B}$ be the set of odd four-digit numbers that have the property that the sum of their digits is even.

The set we are interested in is $\text{A - B.}$ Now $\text{B} \subseteq \text{A}$ and thus

$|\text{A - B}| = |\text{A}| - |\text{B}|$

$= 4500 - 2250$

$= 2250,$

where $|\text{A}| = 4500$  and $|\text{B}| = 2250$

Why $|\text{A}| = 4500?$

There are $9 \cdot 10^2 \cdot 5 = 4500.$ The recipe is as follows:

  1. Choose the first digit (which can’t be $0) - 9$ choices
  2. Choose the second digit – $10$ choices
  3. Choose the third digit – $10$ choices
  4. Choose the fourth digit – $5$ choices (the last digit must be odd – only $5$ choices)


Why $|\text{B}| = 2250?$

There are $5 \cdot 9 \cdot 10 \cdot 5 = 2, 250.$ The recipe is as follows.

  1. Choose the last digit (which must be an odd digit) – $5$ choices
  2. Choose the first digit (which can’t be zero) – $9$ choices
  3. Choose the second digit – $10$ choices
  4. Choose the third digit (two cases) – $5$ choices
    • Case i: If the sum of the other three digits is even, the third digit must be an even digit.
    • Case ii: If the sum of the other three digits is odd, the third digit must be an odd digit
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4 votes
4 votes

Intuitive Solution

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