- #1

Jonathanlikesmath

- 17

- 15

- Homework Statement
- How many 7-digit numbers are even or have exactly three digits equal to 0?

- Relevant Equations
- ## |A \cup B| = |A| + |B| - |A \cap B| ##

A = set of even 7-digit even numbers ## = 9 * 10 * 10 * 10 * 10 * 10 * 5 = 4500000 ##

B = set of 7 -digit numbers with three 0s ## = 9 * { 6 \choose 3} * 9 * 9 * 9 = 131220 ##

## |A \cap B| ##

Portion that does not end in ## 0 = 9 * { 5 \choose 3} * 9 * 9 * 4 = 9^3{ 5 \choose 3}4 = 29160 ##

Portion that does end in ## 0 = 9 * { 5 \choose 2} * 9 *9* 9 * 1 = 9^4 { 5 \choose 2} 1 = 65610 ##

## |A \cup B| = |A| + |B| - |A \cap B| ## = 4500000 + 131220 - (29160 + 65610) = 4536450

My issue is that I "know" I need to use ## { n \choose k} ## in calculating |B| and in both portions of ## |A \cap B| ## but I do not know why I need to use it. From the text, ##{ \textbf{Definition}}## If n and k are integers, then ## {n \choose k} ## denotes the number of subsets that can be made by choosing k elements from an n-element set." I do not see the connection between the definition and how it is applied in the above problem.

Jonathan

B = set of 7 -digit numbers with three 0s ## = 9 * { 6 \choose 3} * 9 * 9 * 9 = 131220 ##

## |A \cap B| ##

Portion that does not end in ## 0 = 9 * { 5 \choose 3} * 9 * 9 * 4 = 9^3{ 5 \choose 3}4 = 29160 ##

Portion that does end in ## 0 = 9 * { 5 \choose 2} * 9 *9* 9 * 1 = 9^4 { 5 \choose 2} 1 = 65610 ##

## |A \cup B| = |A| + |B| - |A \cap B| ## = 4500000 + 131220 - (29160 + 65610) = 4536450

My issue is that I "know" I need to use ## { n \choose k} ## in calculating |B| and in both portions of ## |A \cap B| ## but I do not know why I need to use it. From the text, ##{ \textbf{Definition}}## If n and k are integers, then ## {n \choose k} ## denotes the number of subsets that can be made by choosing k elements from an n-element set." I do not see the connection between the definition and how it is applied in the above problem.

Jonathan