Solution: 2019 Winter Midterm - 2

Author: Michiel Smid

Question

Consider permutations of the set $\{a,b,c,d,e,f,g\}$ that do not contain $bge$ (in this order).
How many such permutations are there?

(a)

$7! - 5!$

(b)

$7! - 4!$

(c)

$7! - 6!$

(d)

$7! - 3!$

A = Permutations that contain $ bge $

We can treat $ bge $ as a single entity.

B = Permutations that do not contain $ bge $

So there’s $ bge, a,c,d,f $ which is 5 entities.

$ |B| = 5! $

$ |S| = $ Total permutations without restrictions

$ |S| = 7! $

$ |\text{Number of permutations that do not contain} bge| = |S| - |B| $

$ |\text{Number of permutations that do not contain} bge| = 7! - 5! $