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# Counting

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105
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How many ways are there to answer a 10-question true/false test, where at least $$8$$ of the questions have been answered with a false?

Apr 15, 2022

#1
+2455
+1

There are 2 options for 1 question

There are another 2 options for the other question

The remaining 8 must all be false.

There are $$10 \choose 8$$ ways to choose which questions are guaranteed to be false

So, in total, there is: $$2 \times 2 \times {10 \choose 8} = \color{brown}\boxed{180}$$ ways to answer.

Apr 15, 2022

#1
+2455
+1

There are 2 options for 1 question

There are another 2 options for the other question

The remaining 8 must all be false.

There are $$10 \choose 8$$ ways to choose which questions are guaranteed to be false

So, in total, there is: $$2 \times 2 \times {10 \choose 8} = \color{brown}\boxed{180}$$ ways to answer.

BuilderBoi Apr 15, 2022
#2
+124598
+2

You are very good at this  counting stuff......I'm usually stuck on many  of these  !!!

CPhill  Apr 15, 2022
#3
+578
+5

That is true,

You seem to get these counting and prob questions with ease. These type of questions always mess up my brain!!!

-Vinculum

Vinculum  Apr 15, 2022
#4
+2455
0

Thank You!!! I grinded AoPS Counting and Probability for a month, and I guess it's paying off now...

BuilderBoi  Apr 15, 2022