SOLUTION: The possible ways to complete a multiple-choice test consisting of 21 questions, with each question having four possible answers (a, b, c, or d).

Question 1197951: The possible ways to complete a multiple-choice test consisting of 21 questions, with each question having four possible answers (a, b, c, or d).

Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

Consider a test with exactly 2 questions, four choices each (a,b,c,d).

We can lay out a 4 by 4 table showing all the possible outcomes. There are 4^2 = 4*4 = 16 outcomes possible.

	a	b	c	d
a	a, a	b, a	c, a	d, a
b	a, b	b, b	c, b	d, b
c	a, c	b, c	c, c	d, c
d	a, d	b, d	c, d	d, d

Question 1's answer choices are marked in red, question 2's answer choices are marked in blue.
Something like "a, d" in the bottom left corner represents selecting 'a' on question 1, and 'd' on question 2.

You do NOT have to write this table into your notes or homework.
This is just something to think about. It hopefully helps provide a visual as to why multiplication occurs.

Extend this idea for 3 questions and we have 4^3 = 4*4*4 = 64 different outcomes. Imagine a 4 by 4 by 4 cube that has volume 64 cubic units.

Unfortunately the visual doesn't extend to 4 questions since it's tricky to think in the fourth dimension. But we still will multiply the terms as before. So there are 4*4*4*4 = 4^4 = 256 different outcomes if there are four questions with four answer choices each.

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In short, if we had 21 questions to this test with four answer choices each, then there are 4^21 = 4,398,046,511,104 different outcomes. This large value is slightly over 4.3 trillion.

Since it's so large, I wouldn't expect my students to write it out. It would honestly be cruel if any teacher expects this level of tedious busy-work. I would stick to the notation 4^21 as it's easier to write and think about.

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Answer: 4^21

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