Discrete Maths: Inclusion Exclusion - patterns “spin”, “game”, “path” or “net”

Question

This is a question from Discrete and combinatorial mathematics by Ralph Grimaldi. This question is related to Inclusion Exclusion Principle.

Question -

Find the number of permutations of a, b, c, ....x, y, z (26 letters) in which none of the patterns “spin”, “game”, “path” or “net” occurs.

Answer given in the book -

N(c¯1 c¯2 c¯3 c¯4) = 26! − [3(23!) + 24!] - (20! + 21!)

(Why does the answer have two consecutive negative signs ? Isn't it supposed to alternate between the terms.)

My solution -

Let c1, c2, c3 and c4 be the set of permutations where “spin”, “game”,
“path”or “net” occurs respectively. Therefore, we need to determine N(¯c1c¯2c¯3c¯4).  
Clearly, N = 26!, 
N(c1) = N(c2) = N(c3) = (26−4+1)! = 23!
N(c4) = (26−3+1)! = 24! 
Similarly, N(c1c2) = N(c1c3) = N(c2c3) = (26 − 8 + 2)! = 20! 
N(c1c4) = N(c2c4) = N(c3c4) = (26 − 7 + 2)! = 21!
N(c1c2c3) = = (26 − 12 + 3)! = 17!
N(c1c2c4) = = (26 − 11 + 3)! = 18!
N(c1c2c3c4) = = (26 − 15 + 4)! = 15!
N(c¯1 c¯2 c¯3 c¯4) = 26! − [3(23!) + 24!] + [3(20!) + 3(21!)) - [17! + 18!] + 15!

I searched on Google and found a solution here

1) I am unable to understand why the terms for N(c1c2c3), N(c1c2c4) and N(c1c2c3c4) are missing in the final answer.

2) Why does the answer have two consecutive negative signs? Isn't it supposed to alternate between the terms. (Look at Grimaldi's solution)

Answer 1

Notations used in the book's answer: $c_1$:"spin", $c_2$:"game", $c_3$:"path", $c_4$:"net".

Let's check all the pairwise combinations. We'll see that some of them have words that can never occur together in a random permutation of the alphabet, so their counts will be zero:

• $c_1c_2$: These can occur together, and were computed to be $N(c_1c_2) = 20!$
• $c_1c_3$: "spin" and "path" both have a "p" that can't be shared.
• $c_1c_4$: These can occur together ("spinet"), and were computed to be $N(c_1c_4) = 21!$
• $c_2c_3$: "game" and "path" both have an "a" that can't be shared.
• $c_2c_4$: "game" and "net" both have an "e" that can't be shared.
• $c_3c_4$: "path" and "net" both have a "t" that can't be shared.

Note that any three, or all four, of the words have at least one letter that cannot be shared, so those will not contribute to the final answer. That's why they're "missing" from the final answer; their counts are zero.

I'm not sure why the last negative is there; I suspect it's a typo. I couldn't find an errata listing for this book, so I don't know for sure, but it looks like the second solution you posted is the correct one.