MATH SOLVE

4 months ago

Q:
# Find the number of five-digit multiples of 5, where all the digits are different, and the second digit (from the left) is odd.

Accepted Solution

A:

Given:

candidates for digits from left:

1. [1-9] (leftmost, 5 digit-number)

2. [13579] odd

3. [0-9]

4. [0-9]

5. [05] (multiple of 5).

We will start with digits with most constraints (5,2,1,3,4)

For the fifth digit, number ends with either 0 or 5, so two cases.

Case 1:

5th digit is a 0 (one choice)

2nd digit can be chosen from [1,3,5,7,9] so 5 choices

1st digit can be chosen from [1-9] less second digit, so 8 choices

3rd digit can be chosen from remaining 7 choices

4th digit can be chosen from remaining 6 choices

for a total of 1*5*8*7*6=1680 arrangements

Case 2

5th digit is a 5 (one choice)

2nd digit can be chosen from [1379] , so 4 choices

1st digit can be chosen from [1-4,6-9] less 2nd digit, so 8-1=7 choices

3rd digit can be chosen from remaining 7 choices

4th digit can be chosen from remaining 6 choices

for a total of 1*4*7*7*6=1176

Case 1+ Case 2

= total number of arrangements for 5 digit multiples of 5 with second digit odd

= 1680+1176

= 2856

candidates for digits from left:

1. [1-9] (leftmost, 5 digit-number)

2. [13579] odd

3. [0-9]

4. [0-9]

5. [05] (multiple of 5).

We will start with digits with most constraints (5,2,1,3,4)

For the fifth digit, number ends with either 0 or 5, so two cases.

Case 1:

5th digit is a 0 (one choice)

2nd digit can be chosen from [1,3,5,7,9] so 5 choices

1st digit can be chosen from [1-9] less second digit, so 8 choices

3rd digit can be chosen from remaining 7 choices

4th digit can be chosen from remaining 6 choices

for a total of 1*5*8*7*6=1680 arrangements

Case 2

5th digit is a 5 (one choice)

2nd digit can be chosen from [1379] , so 4 choices

1st digit can be chosen from [1-4,6-9] less 2nd digit, so 8-1=7 choices

3rd digit can be chosen from remaining 7 choices

4th digit can be chosen from remaining 6 choices

for a total of 1*4*7*7*6=1176

Case 1+ Case 2

= total number of arrangements for 5 digit multiples of 5 with second digit odd

= 1680+1176

= 2856