गणितसारसंग्रह (महावीराचार्य - प्रो. लक्ष्मीचन्द्र जैन सविस्तार हिन्दी अनुवाद सटीक)

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Chapters III and IV are devoted to elementary operations
with fractions. Mahāvīrāchārya has paid considerable a tention to
the problem of expression of a unit fraction as the sum of unit
fraction. This problem has interested mathematicians from remote
antiquity ( Ahmes Papyrus 1650 B. C. ). Here are three relevant
problems ( II 75, 77, 78 ) set in modern notation.
( 1 ) 1 = ½ + ⅓ + 1/3² + ······ + 1/3ⁿ⁻² + 1/(2ⁿ⁻² · 3);
( 2 ) 1 = 1/(2·3·½) + 1/(3·4·½) + ··· + 1/((2n - 1) 2n·½) + 1/(2n·½);
( 3 ) 1/n = a₁ / (n (n + a₁)) + a₂ / ((n + a₁) (n + a₁ + a₂)) + ······

• aᵣ₋₁ / ((n + a₁ + a₂ + ··· + aᵣ₋₂) (n + a₁ + a₂ + ··· + aᵣ₋₁))
• aᵣ / (aᵣ (n + a₁ + a₂ + ··· + aᵣ))
  Problem IV 4: One third of a herd of elephants and three times
  the square root of the remaining part ( of the herd ) were seen on the
  mountain slope; and in a lake was seen a male elephant along with
  three female elephants. How many were the elephants there ?
  Here is a sample of monkish humour !
  Chapter V treats ‘Rule of Three’ and its generalised forms.
  Chapter VI. Having created the arithmetical apparatus in the
  earlier chapters, in this long chapter, Mahāvīrāchārya applies it to
  solving many problems which one encounters in life such as money-
  lending, number of combinations of given things, indeterminate
  equations of first degree, etc.
  Problem (VI 128½) : In relation to twelve ( numerically equal )
  heaps of pomegranates which having been put together and combined
  with five of those ( same fruits ) were distributed equally among 19
  travellers. Give out the numerical measure of ( any ) one heap.
  Problem ( VI 218 ) : The number of combinations of n different
  things taken r at a time is
  n (n - 1) (n - 2)···(n - r + 1) / (1·2·3······r) or n! / (r! (n - r)!)