You must know the story behind offering chhappan Bhogas (56 dishes) to Lord Krishna. According to the story, Sri Krishna held up the Govardhan mountain on his little finger for a week to protect villagers and cattle from a severe downpour caused by the anger of god Indra. He held up the mountain for seven days, without having his regular eight meals a day. When the downpours stopped, the villagers showed their gratitude by offering Him fifty-six different dishes which was the compensation for the eight meals of the day times seven days. Do you see any Math here? Of course, 8 dishes a day times 7 days = 56 dishes.
But, what if Krishna had held the mountain for 10 days? Had villagers managed to offer Him 80 distinct dishes? In that case, Indian cookery must have 80 distinct dishes. Isn’t it? Let us see why there are only 56 dishes.
In Ayurveda , there are six types of tastes (rasas) mentioned. They are;
A. Bitter (katu)
B. Hot or Pungent (Tikta)
C. Astringent (Kashaya)
D. Sour (Amla)
E. Salty (lavana)
F. Sweet (madhura)
A dish is prepared by any possible combination of the listed tastes. In general practice, no dish can be prepared by using only one taste (rasa) or all six tastes (rasas) together. So, the following different combinations can be done (if we do not want to apply combinatorics).
- Taking Two tastes together:
AB, CD, EF, DF, BC, AC, DE, AD, AE, AF , BD, BE, BF, CE, CF
Number of combinations = 15
- Taking Three tastes together:
ABC, ABD, ABE, ABF, ACD, ACE, ACF, ADE, ADF, AEF, BCD,
BDE, BCF, BDF, BCE, BEF, CDE, CDF, CEF, DEF
Number of combinations = 20
- Taking Four tastes together:
ABCD, ABCE, ABCF, ABDE, ABDF, ABEF, ACDE, ACDF, ACEF,
ADEF, BCDE, BCDF, BCEF, BDEF, CDEF
Number of combinations = 15
- Taking Five tastes together:
ABCDE, ABCDF, ABCEF, ACDEF, BCDEF, ABCEF
Number of combinations = 6
Therefore, total number of combinations = 15 + 20 + 15 + 6 = 56
Now, a pure combinatorial version of the situation can be put like this:
Question. How many different dishes can be prepared using six different types of tastes (rasas) if neither of them can be used separately nor all of them together?
We know that the number of ways to select r objects from n different objects $$= n_{C_r}$$ where \[ n_{C_r} = \frac{n!}{r! (n-r)! } \] Therefore, the number of ways to select 2 or more tastes but not all out of 6 distinct tastes \[ = 6_{C_2} + 6_{C_3} + 6_{C_4} + 6_{C_5} = 15 + 20 + 15 + 6 = 56. \]
Another Way of Solving:
Each of the six tastes can be either included in preparing a particular dish or be ignored. So, with each taste only 2 things can be done- included or ignored. Thus, total number of all the possible dishes without caring which is included taste and which is ignored taste \[ = 2^{6} = 64.\]
But, in this 64 possibilities we have counted three possibilities which are not allowed.
They are:
- all the tastes are ignored (any dish can not be prepared) = 1 way
- each taste taken separately (not allowed) = 6 ways
- all the tastes are taken together ( not allowed) = 1 way.
Hare Krishna!
Get Mathactive Sunday in your inbox
A new article every Sunday morning on mathematics, learning and teaching, for parents and teachers. Free, and you can unsubscribe at any time.
Prefer videos, or a short post every day?
1 thought on “Mathematics of Chhappan Bhogas (56 Dishes) in Indian Cookery”
Amazing,
Jai Shree Krishna