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 center of a disk of radius $ r $ and mass $ m $ is attached to a spring of spring constant $ k $, inside a ring of radius $ R>r $ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $ T = \frac{2\pi}{\omega} $. The correct expression for $ \omega $ is ( $ g $ is the acceleration due to gravity): 
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 ________.