Compound P reacts with 3 equivalents of NH2OH, indicating the presence of 3 carbonyl groups, forming oxime Q.
Excess CH3I with KOH reacts with Q to methylate all hydroxyl groups, resulting in compound R.
Compound R reacts with iso-butylmagnesium bromide (Grignard reagent) to add alkyl groups. Hydrolysis with \( \text{H}_3\text{O}^+ \) yields compound S.

\[ P \xrightarrow{\text{3 NH}_2\text{OH}} Q \xrightarrow{\text{CH}_3\text{I}, \text{KOH}} R \xrightarrow{\text{iso-BuMgBr}, \text{H}_3\text{O}^+} S \]
The structure of compound S contains a total of 12 methyl groups:
Total \( -\text{CH}_3 \) groups in compound S: 12
To solve this problem, we need to identify the structure of the organic compound P and follow its transformations to determine the number of methyl groups in compound S. Here's a breakdown of the analysis and reactions involved:
Conclusion: Each branch from original methoxy and appended iso-butyl raises the methyl group count beyond initial methoxy. Therefore, final computation on such rigorous assumptions stalwartly certifies the applicable number of methyl groups in S as 12.
In the given reaction sequence, the structure of Y would be:
The reaction sequence given below is carried out with 16 moles of X. The yield of the major product in each step is given below the product in parentheses. The amount (in grams) of S produced is ____. 
Use: Atomic mass (in amu): H = 1, C = 12, O = 16, Br = 80
Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is
As shown in the figures, a uniform rod $ OO' $ of length $ l $ is hinged at the point $ O $ and held in place vertically between two walls using two massless springs of the same spring constant. The springs are connected at the midpoint and at the top-end $ (O') $ of the rod, as shown in Fig. 1, and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $ f_1 $. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2, and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is $ f_2 $. Ignoring gravity and assuming motion only in the plane of the diagram, the value of $\frac{f_1}{f_2}$ is:
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.