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I have hard time counting the digits

 

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

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

* The sum of the digits is 10

 Jan 29, 2023
 #1
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I would list out the possible number conbiations and then work out how many numbers I can make with each combination.

 

The maximum number of 3s is  3  that is  3331   there are 4 ways to order this

 

Now how many wars can there be two 3s

3,3,2,2                4!/(2!2!) = 6ways to order

3,3,1,1,2             5!/(2!2!)=30 ways to order

3,3,1,1,1,1,         6!/(2!4!) = 15 ways to order 

 

How many ways can there be one 3s

3,2,2,2,1             5!/(3!) = 

3,1,1,2,2,1           6!/(3!2!)=

3,1,1,1,1,2,1            7!/(5!)=

3,1,1,1,1,1,1,1         8!/7! =

 

Ok no more combinations use a 3

now go through a similar process with the 2s

starting with

2,2,2,2,2

 

etc

 

When you have finished add them all together.

 Jan 30, 2023

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