The faces of a cube are to be numbered with integers 1 through 6 in such a way that consecutive numbers are always on adjacent faces (not opposite ones). The face numbered 1 is on top and the face numbered 2 is toward the front. In how many different ways can the remaining faces be numbered?

Total different ways to number the remaining faces is 10.

Step-by-step explanation:

It is given that consecutive numbers are always on adjacent faces (not opposite ones).

The face numbered 1 is on top and the face numbered 2 is toward the front.

Top = 1

Front = 2

The remaining faces are left right bottom back. Total possibilities are shown below.

S. no. 3 4 5 6

(1) Right Back Bottom Left

(2) Right Back Left Bottom

(3) Right Bottom Left Back

(4) Right Bottom Back Left

(5) Bottom Right Back Left

(6) Bottom Left Back Right

(7) Left Back Bottom Right

(8) Left Back Right Bottom

(9) Left Bottom Right Back

(10) Left Bottom Back Right

When Bottom=3, then we can not place 4 on back because doing this 5 and 6 are opposite ones.

Therefore, total different ways to number the remaining faces is 10.