Doves are sold at the rate of 1 for 5 coins, cranes at the rate of 1 for 20 coins and peacocks at the rate of 1 for 30 coins. a certain man was given 500 coins and told to buy exactly 50 birds for the amusement of the King's son. he was also told to spend exactly 500 coins. nothing more, nothing less. how many of each kind of bird could be getting back in order to avoid death?

Price List:

Doves: 5 coins Cranes: 20 coins Peacocks: 30 coins

Total coins to be spent: 500

Total number of birds to be purchased: 50

Let number of doves bought be x, number of cranes bought be y and number of peacocks bought be z

⇒ x + y + z = 50 ... (i)

and 5x + 20y + 30z = 500 ... (ii)

Multiplying (i) by 5 and subtracting from (ii)

5x + 20y + 30z - 5x - 5y - 5z = 500 - 250

⇒ 15y + 25z = 250

⇒ 3y + 5z = 50

Now, the above equation is true for y = 0 & z = 10, for y = 5 & z = 7, y = 10 & z = 4, for y = 15 & z = 1.

For all the above values of y and z, let's determine value of x (using x + y + z = 50):

When y = 0, z = 10, ⇒ x = 40 When y = 5, z = 7, ⇒ x = 38 When y = 10, z = 4, ⇒ x = 36 When y = 15, z = 1, ⇒ x = 34

Hence, the options available for purchase are:

40 doves, 0 cranes and 10 peacocks 38 doves, 5 cranes and 7 peacocks 36 doves, 10 cranes and 4 peacocks 34 doves, 15 cranes and 1 peacock