Question 66
a) The letters $A,B,C$ can be arranged in a row in 6 distinct ways:
$$
A B C,\ A C B,\ B A C,\ B C A,\ C A B,\ C B A
$$
In 3 of them, $A$ appears to the left of $B$:
$$
A B C,\ A C B,\ C A B
$$
Among all 24 distinct arrangements of $A,B,C,D$ in a row, in how many does $A$ appear to the left of $B$?
b) A spider has a sock and a shoe for each of its eight feet. In how many different orders can it put them on, assuming the 8 socks and 8 shoes are all distinct and each sock must be put on before its corresponding shoe?
$$
A B C,\ A C B,\ B A C,\ B C A,\ C A B,\ C B A
$$
In 3 of them, $A$ appears to the left of $B$:
$$
A B C,\ A C B,\ C A B
$$
Among all 24 distinct arrangements of $A,B,C,D$ in a row, in how many does $A$ appear to the left of $B$?
b) A spider has a sock and a shoe for each of its eight feet. In how many different orders can it put them on, assuming the 8 socks and 8 shoes are all distinct and each sock must be put on before its corresponding shoe?