+0

# Counting

-1
5
1
+1791

Find the number of positive integers that satisfy both the following conditions:

- Each digit is a 1 or a 2 or a 3

- The sum of the digits is 5

Feb 13, 2024

#1
+394
+2

A lot of casework, but relatively simple.

We find the digits for each number, find the number of ways to arrange them, to represent the different integers.

Here are the possible cases:

5 =

C1) 1, 1, 1, 1, 1

C2) 1, 1, 1, 2

C3) 1, 2, 2

C4) 1, 1, 3

C5) 2, 3

For C1, there is 1 way to order this.

For C2, there is $$\frac{4!}{3!1!}$$, or 4 ways to order this.

For C3, there is $$\frac{3!}{2!1!}$$, or 3 ways to order this.

For C4, there is $$\frac{3!}{2!1!}$$, or 3 ways to order this.

For C5, there is 2 ways to order this.

There are a total of 1+4+3+3+2 = 13 numbers.

Feb 14, 2024

#1
+394
+2

A lot of casework, but relatively simple.

We find the digits for each number, find the number of ways to arrange them, to represent the different integers.

Here are the possible cases:

5 =

C1) 1, 1, 1, 1, 1

C2) 1, 1, 1, 2

C3) 1, 2, 2

C4) 1, 1, 3

C5) 2, 3

For C1, there is 1 way to order this.

For C2, there is $$\frac{4!}{3!1!}$$, or 4 ways to order this.

For C3, there is $$\frac{3!}{2!1!}$$, or 3 ways to order this.

For C4, there is $$\frac{3!}{2!1!}$$, or 3 ways to order this.

For C5, there is 2 ways to order this.

There are a total of 1+4+3+3+2 = 13 numbers.

hairyberry Feb 14, 2024