Bhaskara II Acharya/Lilavati/Chapter XIII/269

: Chapter $\text {XIII}$. Combination of Digits: $269$

 * How many are the variations of the form of the god Sambhu by the exchange of his $10$ attributes held reciprocally in his several hands:
 * namely, the rope, the elephant's hook, the serpent, the tabor, the skull, the trident, the bedstead, the dagger, the arrow, and the bow:
 * as those of Hari by the exchange of the mace, the discus, the lotus and the conch?

Solution
For Sambhu: $3 \, 628 \, 800$.

For Hari: $24$.

Proof
We have that Sambhu has $10$ objects which may be held in any of his $10$ hands.

Similarly, Hari has $4$ objects which may be held in any of his $4$ hands.

Thus we are asked: how many different possible representations can be made of each of these two gods, differentiating between them by means of which object is held in which hand?

Hence the question is an instance of Number of Permutations.

The number of permutations of $n$ objects from $n$ is given by:
 * $^n P_n = n!$

Here we have that Sambhu has $10$ objects, and so:
 * $10! = 3 \, 628 \, 800$

different representations.

Similarly we have that Hari has $4$ objects, and so:
 * $4! = 24$

different representations.