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Question:

Construct 5-digit numbers from the digits $0, 1, 2, 3, 4$. Repetition and $0$ at the beginning isn't allowed.

a) How many 5-digit numbers can be formed?

b) How many of these are odd?

c) How many of these are divisible by 5?

For a), I got $96$, which is correct, by subtracting the number of numbers that begin with $0$ from $_{5} P_{5}$.

For b) and c), I got completely stuck. The answers to b) and c) are $36$ and $24$ respectively.

Community
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Giorgos
  • 29

3 Answers3

5

Hint: finish the sentences:

  • For $b$: A number is odd if and only if the final digit if the number is odd. Therefore, the constructed number is odd if and only if the final digit is...
  • For $c$: A number is divisible by $5$ if and only if the final digit is $0$ or $5$. Therefore, the constructed number is divisible by $5$ if the final digit is...
5xum
  • 123,496
  • 6
  • 128
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1

Notice, use fill in the blank method

a. number of 5-digit numbers, (zero is not the first/leftmost digit), $$=4\cdot 4\cdot 3\cdot 2\cdot 1=96$$

b. number of odd numbers (each should have last digit odd either $1$ or $3$), $$=3\cdot 3\cdot 2\cdot 1 \cdot 2 =36$$

c. number of numbers divisible by $5$ (each should have $0$ as the last digit), $$=4\cdot 3\cdot 2\cdot 1 \cdot 1=24$$

Harish Chandra Rajpoot
  • 37,450
0

hint

A number is odd then the last digit has to be either $1$ or $3$.

For a number to be divisible by $5$ it has to end with $0$ or $5$.

Anurag A
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