In how many ways the letters in BIOENGINEERING be arranged if B must not be next to the O.

Question

Thanks in advance (is this correct answer 14!-(13!x2!)

Answer 1

easier to see the letters this way:
B
I I I
O
E E E
N N N
G G
R

number of ways without restrictions
= 14!/(3!3!3!2!) = 201,801,600
(do you know why I am dividing by 3!3!3!2! ? )

let's put the BO together, could also be OB
let's treat that pair as one element
number of ways = 2(13!)/(3!3!3!2!) = 28,828,800

number of ways they are apart
= 201,801,600 - 28,828,800
= 172,972,800