SOLUTION: How many ways can the numbers 1, 2, 3, 4 and 5 be placed in a line so that neither 1 nor 5 occupy either the first or the last place in the sequence? (A) 6 (B) 24 (C) 36 (D) 54

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Question 206863: How many ways can the numbers 1, 2, 3, 4 and 5 be placed in a line so that neither 1 nor 5 occupy either the first or the last place in the sequence?
(A) 6 (B) 24 (C) 36 (D) 54 (E) 72
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
How many ways can the numbers 1, 2, 3, 4 and 5 be placed in a line so that neither 1 nor 5 occupy either the first or the last place in the sequence? 

We can choose the 1st number 3 ways.  That's 3.
We can choose the 5th number 2 ways.  That's 3*2.
We can choose the 2nd number 3 ways (we've used two already).  That's 3*2*3
We can choose the 3rd number 2 ways.  That's 3*2*3*2
We can choose the 4th number 1 way.  That's 3*2*3*2*1

Multiply that out and get 36. That's 36 ways.  Here they all are:

 1.  2, 1, 3, 5, 4
 2.  2, 1, 4, 5, 3
 3.  2, 1, 5, 3, 4
 4.  2, 1, 5, 4, 3
 5.  2, 3, 1, 5, 4
 6.  2, 3, 5, 1, 4
 7.  2, 4, 1, 5, 3 
 8.  2, 4, 5, 1, 3
 9.  2, 5, 1, 3, 4
10.  2, 5, 1, 4, 3
11.  2, 5, 3, 1, 4
12.  2, 5, 4, 1, 3
13.  3, 1, 2, 5, 4
14.  3, 1, 4, 5, 2
15.  3, 1, 5, 2, 4
16.  3, 1, 5, 4, 2
17.  3, 2, 1, 5, 4
18.  3, 2, 5, 1, 4
19.  3, 4, 1, 5, 2
20.  3, 4, 5, 1, 2
21.  3, 5, 1, 2, 4
22.  3, 5, 1, 4, 2
23.  3, 5, 2, 1, 4
24.  3, 5, 4, 1, 2
25.  4, 1, 2, 5, 3
26.  4, 1, 3, 5, 2
27.  4, 1, 5, 2, 3
28.  4, 1, 5, 3, 2
29.  4, 2, 1, 5, 3
30.  4, 2, 5, 1, 3
31.  4, 3, 1, 5, 2
32.  4, 3, 5, 1, 2
33.  4, 5, 1, 2, 3
34.  4, 5, 1, 3, 2
35.  4, 5, 2, 1, 3
36.  4, 5, 3, 1, 2

Edwin
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