Combinations and permutations please help

Question

Omg pls I'm so stressed right now and I'm hoping this site will help with my math about permutations and combinations, so basically for my assignment I was given a story about Tom and jerry and there 3 problems that I can't find the solution to, if you guys could help that would mean the world to me thank you.

Tom gets back home but wants to take revenge from jerry, because at end of the day Jerry is a mouse and he is higher at the food chain, now this was not just revenge but his struugle for his pride. He plans a trap for Jerry and invites 100 other cats for help, tom forgot that it was just a mouse not Osama bin laden. Out of 100, 50 brought a mele weapon, 30 brought guns, 25 brought rocket launchers, 3 cats brought all the three, 10 brought mele+gun, 5 brought gun+ RPG and 4 brought RPG+ mele. How many cats came empty handed.

This group of 100 cats is also known as Cats Against Terrorism or CAT, CAT had apples, milk, cheese, cake, pie, ramen and vegies for lure, there are how many combinations of lure used if at least two items have to be used.

Were able too capture jerry, they wanted to tease jerry but only five cats who gets elected as president, vice president, secretary, head of cat reltions and CAT ambassador can only get to tease jerry, there are how mny ways in which this body can be elected, tom has to be one on the committee, sure shot

Answer 1

I'll admit that I'm not sure about the first one, but I can definitely help you on the last two and then maybe someone else will be kind enough to help you on the first.

 

2) This is asking for the number of combinations of 2 items from the group of 7. Note that it's combinations and not permutations since order does not matter (i.e. a group of milk & cake is the same thing as cake & milk; keep this is mind for part 3).

 

7C2

=7!/(2! * 5!)

= 7 * 6 / 2

=21

 

3) This one is asking for the number of permutations since order does matter (i.e. if Tom is President and Jerry is VP, then that is not the same thing as Jerry as President and Tom as VP).  While it may seem intuitive to think you are picking permutations of 5 from your group of 100, since Tom has to have a spot guaranteed we are actually picking 4 out of 99.

 

99P4

= 99!/95!

= 90,345,024

 

However, we are still not done here.  That 90 million number is the number of perms of the other 99 cats, but we also have to account for the fact that Tom could be in any of the 5 cabinet spots.  So the final answer is actually:

 

5P1 * 99P4

= 5 * 90,345,024

= 451,725,120