An enormous diversity of problems can be solved by creatively applying algebraic techniques. This quiz showcases some of that diversity, pulling problems from five of the topics that we’ll cover in depth in this Exploration:

- Factorials
- Rates & Ratios
- Sequences and Series
- Proving Algebraic Identities
- Avoid Proving that 1=2

**Factorials**

\[ \large \frac {{\color{red}6!} \times \color{blue}7!}{\color{purple}{10}!} = \, \color{green}? \]

Note: The exclamation points are all factorials: \[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1.\]

**Rates & Ratios**

It takes 4 cooks 4 hours to bake 4 cakes. Assuming no change of rate, how many hours will it take 8 cooks to bake 8 cakes?

**Sequences and Series**

**Proving Algebraic Identities**

What is the sum of the first 13 positive odd numbers?

A big hint:

**Avoid Proving 1=2**

Let \( a = b \).

Then,
\[ \begin{array} {c r c l }

\color{red} {1} & ab & = & a^2 \\
\color{orange} {2} & ab - b^2 & = & a^2 -b^2 \\
\color{gold} {3} & b(a-b) & = & (a+b)(a-b) \\
\color{green} {4} & b & = & a+b \\
\color{blue} {5} & b & = & b+b \\
\color{indigo} {6} & b & = & 2b \\
\color{purple} {7} & 1 & = & 2.
\end{array} \]

What is the **first flawed step** in this proof?

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