In which year did Charu begin her training under Pandit Meghnath?

From clue (4), Ananya starts under $M$ in 2013 and Bhaskar starts under $S$ in 2013, so those two cells are filled.

From clue (2), $R$ can only operate in 2013-2014 and in 2019-2020 — two separate blocks of 2 years each.

Since a Guru must train for consecutive years, $R$'s span cannot exceed $2$, so $R$ has span $2$.

Therefore $M$ and $S$ have spans $3$ and $4$ in some order.

Each musician needs $R$ for 2 years inside one window (2013-2014 or 2019-2020), and each window holds at most 2 of them. With 4 musicians and 2 windows, each window takes exactly 2.

Ananya is under $M$ across 2013-2014, so she cannot be under $R$ then; her $R$ must be 2019-2020.

Bhaskar is under $S$ across 2013-2014, so his $R$ must also be 2019-2020.

Thus the 2019-2020 window is $\{Ananya, Bhaskar\}$, forcing $\{Charu, Devendra\}$ into 2013-2014.

This fits clue (3): the forbidden pairs Ananya-Devendra and Bhaskar-Charu land in different windows, while Charu-Devendra become Gurubhai for 2 years.

By clue (3) Ananya and Charu are Gurubhai for 2 years. This cannot be under $R$ (different windows) nor under $S$ (clue 1 allows only one student under $S$ at a time), so it must be under $M$.

Ananya is under $M$ from 2013; Charu's $M$ can only start in 2015 (she is under $R$ in 2013-2014).

If $M$ had span $3$, Ananya's $M = 2013\text{-}2015$ and Charu's earliest $M = 2015\text{-}2017$ overlap only in 2015 — just 1 year.

So $M$ has span $4$: Ananya's $M = 2013\text{-}2016$ and Charu's $M = 2015\text{-}2018$ overlap in 2015-2016, exactly 2 years.

Hence $M$ span $=4$ and $S$ span $=3$. Filled accordingly.

Bhaskar's $M$ has span $4$. He is under $S$ in 2013-2015 (started 2013, span 3) and under $R$ in 2019-2020.

The only remaining block of 4 consecutive free years is 2021-2024.

So Bhaskar's $M = 2021\text{-}2024$.

By clue (3) Bhaskar and Devendra are Gurubhai for 2 years, which must be under $M$. Devendra's $M$ (span 4) must overlap Bhaskar's $M$ (2021-2024) in exactly 2 years.

Devendra is under $R$ in 2013-2014, leaving 2019-2020 free. Checking the windows: $2019\text{-}2022$ overlaps $2021\text{-}2024$ in exactly 2 years (2021-2022), whereas $2020\text{-}2023$ gives 3 and $2018\text{-}2021$ gives 1.

So Devendra's $M = 2019\text{-}2022$.

$S$ has span $3$, and by clue (1) at most one musician is under $S$ in any year, so all four $S$-periods are disjoint.

Bhaskar's $S = 2013\text{-}2015$ is filled.

Devendra's free 3-year blocks for $S$ are 2015-2017 or 2016-2018; the year 2015 is already taken by Bhaskar's $S$, so Devendra's $S = 2016\text{-}2018$.

Ananya's remaining 3-year $S$ options are $2021\text{-}2023$ or $2022\text{-}2024$; Charu's lie within 2019-2024. Their $S$-periods must be disjoint (clue 1).

If Ananya's $S = 2021\text{-}2023$, every 3-year block for Charu overlaps it, leaving no slot. So Ananya's $S = 2022\text{-}2024$, and Charu's $S = 2019\text{-}2021$.

The remaining blank cells are gap years: Ananya (2017, 2018, 2021); Bhaskar (2016, 2017, 2018); Charu (2022, 2023, 2024); Devendra (2015, 2023, 2024).

Gurubhai check: Charu-Devendra under $R$ in 2013-2014; Ananya-Charu under $M$ in 2015-2016; Ananya-Bhaskar under $R$ in 2019-2020; Bhaskar-Devendra under $M$ in 2021-2022. Each of these four pairs is Gurubhai exactly 2 years, and Ananya-Devendra and Bhaskar-Charu never share a Guru. This matches clue (3); $S$ has exactly one student each year, and no Guru ever has more than two. The arrangement is unique.

Charu trains under Pandit Meghnath ($M$) during 2015-2018, beginning in 2015. Among the options (2015, 2021, 2017, 2016), the answer is 2015 (option 1).