Is there a simple way to explain why ISRO's PSLV launch vehicle uses four stages to get to LEO, and why they alternate as follows:
I am sure the optimization is a bit complicated, but there may be a simple way to explain how this four stages to LEO worked out best for them in the early 1990's.
I just watched the video below; PSLV is briefly described and each stage shown separately starting at about 02:00.
PSLV is a bit of a weird duck because it's a transitional step from small solid-rocket designs to larger ones that rely more heavily on liquid rocket engines with higher specific impulse. It doesn't represent an uncompromised "clean sheet" optimal design.
ISRO's first two orbital launcher designs, SLV (1979-1983) and ASLV (1987-1994), were small, not-very-successful 4-stage and 5-stage solid rocket designs lofting very small payloads. Solids are cheap and simple, scalable up to a point, but not very mass-efficient; they require fairly heavy casings which penalize them more than their specific impulse would indicate.
PSLV is much larger -- 7 times the liftoff mass of ASLV. ISRO's solid stages have good specific impulse as solids go, but liquid engines offered potentially better performance. However, ISRO didn't have big liquid rocket engines available -- a couple of Indian attempts to develop liquid rocket engines for military missiles (Devil, Valiant) in the 1970s were generally unsuccessful.
They licensed the Viking engine used on the Ariane series, modifying it as the Vikas; this wasn't big enough to be a good first-stage single engine but was a reasonable choice for a second stage. A hypergolic liquid final stage had the advantage of better control of final orbit insertion and restartability (PSLV has been a prodigious multi-satellite launcher), and the engines used here are the same type used for roll control of the first stage, so didn't require an additional development program. The use of two liquid stages increased the payload to LEO from ASLV's 150 kg to 3800 kg - a 25x increase from a 7x heavier rocket!
As addressed in another QA, PSLV's successor GSLV retains the big solid first-stage, with Vikas-engined strap-on liquid boosters that actually burn longer than the core -- another transitional compromise based on the available components -- and a new upper stage using a hydrogen-oxygen engine (originally the Russian KVD-1, then the indigenous derivative CE-7.5) with much better specific impulse.
The newest ISRO launcher, GSLV-III, is a much more "modern conventional" arrangement: solid boosters augmenting a hypergolic liquid first stage (Vikas again, paired this time) for cheap high thrust; hydrogen-oxygen upper stage for best specific impulse.
