How many numbers with three distinct digits have digits from {1,2,3,4,5} and a digit is 3?

1 Answer
MathFact-orials.blogspot.com
Aug 28, 2017

36

Explanation:

We have the numbers #1, 2, 3, 4, 5# to choose from to make 3 digit numbers and one of those numbers needs to be a 3. That means we can look at this problem this way - of the three digits, we can put a 3 in the first digit:

#3XX#

leaving four digits to put into the #X# places. Since this will also be true with the 3 here:

#X3X#

and here

#XX3#,

we can then say that whatever result we get for the first option we can then multiply by 3 to find all the options.

As far as choosing the remaining two numbers , we can either find that directly:

#4xx3=12#

or go via the permutation route (permutation because we care about order) to get to:

#P_(n,k)=(n!)/((n-k)!); n="population", k="picks"#

#P_(4,2)=(4!)/((4-2)!)=(4!)/(2!)=(4xx3xx2!)/(2!)=4xx3#

Putting it all together, we get:

#(4xx3)xx3=36#

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