**Questions to ponder: (detailed solutions below)**

**#1: Camy made a list of every possible distinct four-digit positive integer that can be formed using each of the digits 1, 2 , 3 and 4 exactly once in each integer. What is the sum of the integers on Camy's list?**

**#2: Camy made a list of every possible distinct five-digit positive even integer that can be formed using each of the digits 1, 3, 4, 5 and 9 exactly once in each integer. What is the sum of the integers on Camy's list? (2004 Mathcounts Chapter Sprint #29)**

Solutions:

**#1:**

**Solution I:**

There are 4! = 24 ways to arrange the four digits. Since each digit appears evenly so each number will appear 24 / 4 = 6 times.

1 + 2 + 3 + 4 = 10 and 10 times 6 = 60 ; 60 (1000 +100 +10 + 1) = 60 x 1111 =

**66660**, which is the answer.

**Solution II:**

the four numbers.

2.5 (1000 + 100 + 10 + 1) x 24 =

**66660**

**#2:**

**Solution I:**

Since this time Camy wants five-digit

**even**integer, which means that the number "4" has to be at the unit digit and only 1, 3, 5, 9 can be moved freely.

Again there are 4! = 24 ways to arrange the four numbers. 1 + 3 + 5 + 9 = 18 and 18 x 6 = 108 (Each number that can be moved freely appears 6 times evenly.)108 x 11110 + 4 x 24 =

**1199976**

**Solution II:**

Since this time Camy wants five-digit

**even**integer, which means that the number "4" has to be at the unit digit and only 1, 3, 5, 9 can be moved freely.

There will be 4! = 24 times the even number 4 will be used so 4 x 24 = 96

As for the remaining 4 numbers, their average (or mean) is

4.5

***( 10000 + 1000 + 100 + 10) * 24 (arrangements) + 96 = 4.5 * 11110 * 24 + 96 =

**1199976**

**Other applicable problems: (answers below)**

**#1: What is the sum of all the four-digit positive integers that can be written with the digits 1, 2, 3, 4 if each digit must be used exactly once in each four-digit positive integer? (2003 Mathcounts Sprint #30)**

**#2: What is the average (mean) of all 5-digit numbers that can be formed by using each of the digits 1, 3, 5, 7, and 8 exactly once? (You can use a calculator for this question.) (2005 AMC-10 B)**

**#3: What is the sum of all the four-digit positive integers that can be written with the digits 2, 4, 6, 8 if each digit must be used exactly once in each four-digit positive integer?**

**#4: What is the sum of all the 5-digit positive odd integers that can be written with the digits 2, 4, 6, 8, and 3 if each digit must be used exactly once in each five-digit positive integer?**

**#5:**

**What is the sum of all the four-digit positive integers that can be written with the digits 2, 3, 4, 5 if each digit must be used exactly once in each four-digit positive integer?**

**Answer key:**

**#1: 66660**

**#2:**

**#3: 133320**

**#4: 1333272**

**#5: 93324**

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