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Home » Given $\cos \theta = \frac13$, compute: - Midis

Given $\cos \theta = \frac13$, compute: - Midis

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 This is given by:

📌 \binom{8 - 3 + 1}{3} = \binom{6}{3} = 20

📌 This formula counts the number of ways to place $k$ non-adjacent indistinct items in $n$ positions.

📌 So, there are 20 ways to choose positions for the 3 A’s such that no two are adjacent.

📌 Step 2: Assign labels to the remaining 5 positions.

📌 Each of the 5 non-A positions can be labeled U, C, or G (3 choices each). So, number of labelings is $3^5 = 243$.

📌 Therefore, total number of valid sequences is:

📌 \times 243 = 4860

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Now use $ p(1) $: $ 0 + 2 + 1 + d = 3 \Rightarrow d = 0 $.
Thus, $ p(x) = 2x^2 + x $. Then $ p(0) = 0 $.
But wait—this is quadratic, but the problem states $ p(x) $ is cubic....
Wait—perhaps the problem allows degree ≤ 3. Many contest problems phra...
But let's verify all values:
$ p(1) = 2 + 1 = 3 $,
$ p(2) = 8 + 2 = 10 $,
$ p(3) = 18 + 3 = 21 $,
$ p(4) = 32 + 4 = 36 $.
All match. Thus, even if not strictly cubic, $ p(0) = 0 $.

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