In how many different ways can 7 boys and 7 girls be seated alternatively in a row from left to right?

B: a boy
G: a girl

Firstly, arrange the 7 boys in a row as _B_B_B_B_B_B_B_ (₇P₇).
Then, place the 7 girls in the leftmost 7 spaces (i.e. GBGBGBGBGBGBGB_) (₇P₇) OR the rightmost 7 spaces (i.e. _BGBGBGBGBGBGBG) (₇P₇).

Number of different ways
= ₇P₇ × (₇P₇ + ₇P₇)
= ₇P₇ × ₇P₇ × 2
= 5,040 × 5,040 × 2
= 50,803,200

There are no restrictions on which boy sits next to which girl. This is the same as asking:

Youth occupy two rows in a church, boys in one and girls in another. How many different ways to seat them?

First you have 2 choices: who sits in the front row. Then for each row you have 7! ways to seat each sex. 

Thus 2*7!*7! in all. 

Order matters, because individuals are not interchangeable.

7 boys to choose from * 7 girls to choose from * 6 boys to choose from * 6 girls to choose from * .... * 1 boy to choose from * 1 girl to choose from =>
7! * 7!

However, we can also start with 7 girls * 7 boys * 6 girls * 6 boys * ... * 1 girl * 1 boy

7! * 7! + 7! * 7! =>
2 * 7! * 7! =>
2 * 5040 * 5040 =>
10080 * 5040 =>
(10000 + 80) * (5000 + 40) =>
50,000,000 + 400,000 + 400,000 + 3,200 =>
50,803,200