Question

1. tatiana has a special puzzle in which all of the pieces fit together in any way. there is no goal picture. instead, the goal of the puzzle is to make different patterns and pictures using the pieces. if tatiana has 50 unique puzzle pieces and she plans to use all of them, how many possible pictures can she create?

options:
\\(\frac{50!}{0!}\\)
\\(50^{50}\\)
\\(\frac{50!}{50!}\\)
\\(\frac{99!}{(50! \times 49!)}\\)

Explanation:

Step1: Identify the problem type

This is a permutation problem where we need to find the number of ways to arrange 50 unique puzzle pieces (using all of them). The formula for permutations of \( n \) distinct objects is \( P(n, n)=\frac{n!}{(n - n)!}=\frac{n!}{0!} \), and we know that \( 0!=1 \).

Step2: Apply the permutation formula

Here, \( n = 50 \). So the number of permutations (possible pictures) is \( \frac{50!}{0!} \) (since \( 0!=1 \), this is equivalent to \( 50! \), but the form \( \frac{50!}{0!} \) matches one of the options).

Let's analyze the other options:

• \( 50^{50} \): This would be the case if for each of the 50 positions, we have 50 choices, but that's not the case here as we are arranging distinct pieces (no repetition in the sense of using each piece once).
• \( \frac{50!}{50!}=1 \): This is not correct as it would imply only one way, which is wrong.
• \( \frac{99!}{(50!\times49!)} \): This is the combination formula \( C(99, 50) \), which is for choosing 50 items from 99, not relevant here.

Answer:

\( \boldsymbol{\frac{50!}{0!}} \) (the first option)