Another Combination problem

[Mahmoud Ibrahim]

Will you please solve the question of permutations and combinations :
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How many numbers of value less than 400 can be formed from the digits {1, 2, 3, 4, 5} in each of the following.
First: the digits can be repeated
Second: the digits can not be repeated.
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The book said that the answers are : first = 61 . second = 105 .
How can we get to this answer.
Thank you very much.

[shyamjayakannan]

Here, [tex]^nP_r=\frac{n!}{(n-r)!}[/tex]

a) Single digit: all permutations= [tex]5[/tex], double digit: all permutations= [tex]5^2[/tex], triple digit: all permutations - those starting with 4 and 5 = [tex]5^3-2\times5^2[/tex].
Adding everything, we get [tex]5+25+125-2\times25=\boxed{105}[/tex]

b) Single digit: all permutations= [tex]^5P_1[/tex], double digit: all permutations= [tex]^5P_2[/tex], triple digit: all permutations - those starting with 4 and 5 = [tex]^5P_3-2\times{}^4P_2[/tex].
Adding everything, we get [tex]5+20+60-2\times12=\boxed{61}[/tex]