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Now, I have been trying to find the answer both on my own and using the internet for about half an hour, but without success.

I'm being asked the following question:

How many arrangements can be made using the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ which result in a number starting with $a)~4, ~b) ~13, ~c) ~528, ~d)~ 123 $

I'm sure, I'm overthinking things. I'd really appreciate an answer, have been coming back to this question throughout the whole day and still have not been able to come up with a correct solution.

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    Do you have to use all the digits for each arrangement? – 2017-01-30

1 Answers 1

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If repetition is not allowed and all digits must be used

(a) fix $4$ at first and arrange remaining $8$ characters. so $8!$ ways.

(b) fix $1,3$ at first,second and arrange remaining $7$ characters. so $7!$ ways.

similarly, $6!$ ways for (c) and (d)

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    @BrianTung typo, corrected. thanks – 2017-01-30
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    I knew I was severely overthinking this. Thank you so much! These align with my answer sheet(which are obviously without the steps(thanks, teacher))! :) I will accept this answer in 10 minutes(rules). – 2017-01-30
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    glad that I could help. – 2017-01-30