Question 831148
a) a insists on sitting with b.
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That's the number of arrangements of 
the four things {ab, c, d, e}, which is 4! or 24. 
PLUS
the number of arrangements of 
the four things {ba, c, d, e}, which is also 4! or 24.

Answer = 2×24 = 48 ways.
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b) c refuses to sit with d.
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First we find the number of ways anybody can sit anywhere.
That's the number of arrangements of 
the four things {a, b, c, d, e}, which is 5! or 120.

The we subtract from that 5! the number of unwanted arrangements.
The unwanted arrangements are 

The number of arrangements of 
the four things {a, b, cd, e}, which is 4! or 24. 
PLUS
he number of arrangements of 
the four things {a, b, dc, e}, which is also 4! or 24.

as that's 2×24 just as in part a), 

[so we could have just observed that the number of unwanted
arrangements here was the same as the answer to a), 48]

Answer: 5! - 2×4! = 120 = 2×24 = 120 - 48 = 72

Edwin</pre>