There is no fixed 'rule' to the number or type of veggies that go into the Bhaji. You can add or substitute any of the veggies with whichever ones you like. It is normally very hard to distinguish what veggies go into this dish when you have it, as everything is normally mashed together.
You can either boil all the veggies in a pan or pressure cooker, mash them, then add them to a tomato based masala...OR...stir fry all the veggies one at a time then let them simmer together until soft and mash the lot once they are all done. I prefer stir-frying as I find this helps to enhance all the flavours of the veggies better than boiling does.
This is my favorite method of preparation:
Heat the oil and butter in a pan or wok, preferably non-stick, then add the cumin seeds and let them splutter. Then add the fenugreek leaves followed by the onions. Let the onions fry until golden, then add the ginger and garlic pastes. Stir fry for a minute, then add the veggies one at a time, stir-frying each for a minute before the next addition. Do them in this order:
Start with the cabbage, followed by the capsicum, carrots, eggplants, cauliflower, potatoes and tomatoes. Reserve the peas for later.
Add salt and all the powdered spices except for the garam masala. Stir fry for a few minutes, then add the green chilies and grated mango. Keep the heat on low keep stirring. Add the tomato paste, then add some water (enough to cover the veggies. Cover the pan and let the veggies cook until the water reduces and the veggies are soft. This could take about 20 minutes. Make sure you keep checking on the veggies and giving them a stir every now and then, and add water if you find it getting too dry.
Once veggies are soft, mash them using a masher. Then adjust salt and water. Add a few drops of red coloring as needed, just to give the mixture a lovely fresh color. Simmer for a few minutes then add the peas and give them a stir. Cook for 5 more minutes or until desired consistency is reached, then add the chopped coriander and a tablespoon of butter. Sprinkle the garam masala, stir and switch off the heat.
Just before serving, squeeze the lemon juice, stir it in, garnish with some chopped onions and coriander and serve with the buttered pav and kachumbar made of sliced onions, chopped tomatoes and chopped cucumber.