CAN YOU PLEASE SLOVE THE PROOF AND DON'T COPY PASTE WRONG ANSWERS FROM OTHER CHEGG MEMBERS

Alice is bored and decided to play the following game. She has a regular deck of 52 different cards that she placed on her desk face up and next to each other. She also has a program that simulates drawing a card at random from a similar deck, more precisely, each time she uses the program it shows one of the 52 possible cards with probability 1/52.

Alice then proceeds to virtually draw the first card in the program, and she puts a chip on the corresponding card on her desk.

From the second draw onward, she virtually draws a card in the program and looks at her desk. If there is no chip over the corresponding card on her desk, she places a chip on it; otherwise she just draws a new card. Alice continues playing until she places a chip on the last card on her desk.

Consider the following example in which the first 5 rounds of a game are de- scribed.

1st: Alice draws the 1st card in her program and it is a 10 , Alice places a chip on the 10 in her desk;

2nd: Alice draws the 2nd card in her program and it is a A , Alice places a chip on the A in her desk;

3rd: Alice draws the 3rd card in her program and it is a 2 , Alice places a chip on the 2 in her desk;

4th: Alice draws the 4th card in her program and it is a A , Alice notices that there is already a chip on the A in her table, thus no chip is placed in this round;

5th: Alice draws the 5th card in her program and it is a Q , Alice places a chip on the Q on her desk.

Let denote the random variable that counts the number of cards Alice has virtually drawn with her program. Prove that () satisfies the following in- equalities:

ln() () (ln() + 1),

with = 52.