# 100 tickets numbered 1,2,3…,100, are sold to 100 different people for a drawing…

I found the answer to these questions but I want someone could give explaination

$100$ tickets numbered $1,2,3...,100$ are sold to $100$ different people for a drawing. Four different prizes are awarded, including a grand prize (trip to Tahiti). How many ways are there to award the prizes if:
a) There are no restrictions? Answer: $94109400$
b) The person holding ticket 47 wins the grand prize? Answer: $941094$
c) The person holding ticket 47 wins one of the prizes? Answer: $3764376$
d) The person holding ticket 47 does not win a prize? Answer: $90345024$
e) The person holding ticket 19 and 47 both wins prize? Answer: $114072$

a) If $k$ different prizes are there for $n$ people and no person can get more than one prize then there are: $$n\times(n-1)\times\cdots\times(n-k+1)$$ ways if there are no restrictions.

Do you understand why? See the answer of @ndruiven.

This leads to $100\times99\times98\times97=94109400$ ways in your case where $n=100$ and $k=4$.

b) After handing out the grand prize there are $k=3$ different prizes left for $n=99$ persons. Apply the formula.

c) $4$ times case b) since there are $4$ possibilities when it comes to the prizes that can be won by person $47$.

d) $k=4$ and $n=99$. Person $47$ is left out.

e) First hand out a prize to $47$. There are $4$ possibilities. Then hand out a prize to $19$. There are $3$ possibilities. Then the other prizes are handed out: $k=2$ and $n=98$. Finally you come to $4\times3\times98\times97=114072$ possibilities.

The solution to the first problem is found by using the multiplication principle. There are 4 prizes to give out. How many people can get the first prize? Now that one person has won, how many people can win the second prize? That is now two people who can't win the third prize. How many people can win the third, the fourth? Each successive prize has a different number of possible winners. The total number of possible ways the prizes can be given out is the product of the four numbers you found above. That is,

$P(100,4) = 100 \cdot 99 \cdot 98 \cdot 97 = 94,109,400$.

See if you can figure out the rest.