By: Tao Steven Zheng (郑涛)

【Problem】

Problem 1: Lord Shiva (Bhaskara's ''Lilavati'', c. 1150 AD)

The God Shiva had ten items (one item in each hand): the rope, the hook, the snake, the drum, the skull, the trident, the bedstead, the dagger, the arrow, and the bow. How many different ways can these items be exchanged?

Problem 2: Six Flavour (Mahavira's ''Ganita-Sara-Sangraha'', c. 850 AD)

There are six flavours: astringent, bitter, sour, spicy, salty, and sweet. In how many ways can different flavours be used in combination?

【Solution】

Problem 1: Lord Shiva (Bhaskara's ''Lilavati'', c. 1150 AD)

This is a permutation problem. The number of possible arrangements of Shiva’s items is . This is more compactly written using the factorial symbol

Problem 2: Six Flavour (Mahavira's ''Ganita-Sara-Sangraha'', c. 850 AD)

This problem is the sum of combinations

where

Shortcut
There is a known property of combinations

Using this property gives