Question

Jonathan And Sibo enter a race of 10 competitors.If there are 10 lanes,calculate the number of possible arrangement if:
1)Jonathan and sibo are placed next to each other?
2)Jonathan is placed in the first lane and Sibo is not placed next to Jonathan?

May I please ask how do I answer these questions?i do not know how to answer them at all,please guide me and show me in simple steps and what I should look out for?I don’t know how to do counting principle,please help me.
Thanks

Answer 1

1) If Jonathan and Sibo are placed next to each other we may consider this pair as one element. Then there are $10-1=9$

elements. The number of possible arrangements

$$9!=1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9=362880$$

There are 2 arrangements in pair Jonathan and Sibo.

Hence the total number of arrangements is

$$2\cdot362880=725760$$

2) If Jonathan is placed in the first lane, then Sibo can be placed in the third, fourth, ..., tenth lane:

$$n-2$$

If $n=10$

$$10-2=8$$

There are 8 other competitors.

The number of possible arrangements is

Hence the total number of arrangements is

$$8\cdot8!=322560$$