Difference between revisions of "2024 AMC 12B Problems/Problem 17"

Latest revision as of 19:39, 21 November 2024

Problem 17

Integers $a$ and $b$ are randomly chosen without replacement from the set of integers with absolute value not exceeding $10$ . What is the probability that the polynomial $x^3 + ax^2 + bx + 6$ has $3$ distinct integer roots?

$\textbf{(A) } \frac{1}{240} \qquad \textbf{(B) } \frac{1}{221} \qquad \textbf{(C) } \frac{1}{105} \qquad \textbf{(D) } \frac{1}{84} \qquad \textbf{(E) } \frac{1}{63}$ .

Solution

Solution 1

Since $-10 \le a,b \le 10$ , there are 21 integers to choose from, and $P(21,2) = 21 \times 20 = 420$ equally likely ordered pairs $(a,b)$ .

Applying Vieta's formulas,

$x_1 \cdot x_2 \cdot x_3 = -6$

$x_1 + x_2+ x_3 = -a$

$x_1 \cdot x_2 + x_1 \cdot x_3 + x_3 \cdot x_2 = b$

Cases:

(1) $(x_1,x_2,x_3) = (-1,1,6) , b = -1, a=-6$ valid

(2) $(x_1,x_2,x_3) = ( 1,2,-3) , b = -7, a=0$ valid

(3) $(x_1,x_2,x_3) = (1,-2,3) , b = -5, a=-2$ valid

(4) $(x_1,x_2,x_3) = (-1,2,3) , b = 1, a=-4$ valid

(5) $(x_1,x_2,x_3) = (-1,-2,-3) , b = 11$ invalid

the total event space is $21 \cdot (21- 1)$ (choice of select a times choice of selecting b given no-replacement)

hence, our answer is $\frac{4}{21 \cdot 20} = \boxed{\textbf{(C) }\frac{1}{105}}$

~luckuso

Solution 1.1 (desperation)

As obtained in Solution 1, we get that there are $P(21,2) = 420$ equally likely ordered pairs $(a,b)$ , which means that the denominator will likely be a factor of 420, which leaves answers C and D, and if you are lucky enough, you can guess that the answer is $\boxed{\textbf{(C) }\frac{1}{105}}$ ~Soupboy0

Video Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=ptFW2866-Xw

@@ Line 38: / Line 38: @@
 ==Solution 1.1 (desperation)==
 As obtained in Solution 1, we get that there are <math>P(21,2) = 420</math> equally likely ordered pairs <math>(a,b)</math>, which means that the denominator will likely be a factor of 420, which leaves answers C and D, and if you are lucky enough, you can guess that the answer is <math>\boxed{\textbf{(C) }\frac{1}{105}}</math>
+~Soupboy0
 ==Video Solution 1 by SpreadTheMathLove==
 https://www.youtube.com/watch?v=ptFW2866-Xw
-~Soupboy0
 ==See also==
 {{AMC12 box|year=2024|ab=B|num-b=16|num-a=18}}
 {{MAA Notice}}

2024 AMC 12B (Problems • Answer Key • Resources)
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