Let the base of the number system used in mars be

*n*.

∴ (15)

_{n}, (40)

_{n}, (

*k*01)

_{n} and (122)

_{n} are in arithmetic progression.

When expressed in the decimal system, these numbers will be (5 +

*n*), 4

*n*, (1 +

*kn*^{2}) and (

*n*^{2} + 2

*n* + 2), respectively.

If the numbers are in AP in base

*n* then their corresponding values in base 10 will also be in AP.

∴ (5 +

*n*), 4

*n*, (1 +

*kn*^{2}) and (

*n*^{2} + 2

*n* + 2) are in arithmetic progression in the decimal number system.

Since in a AP, the difference between any two consecutive terms is equal,

∴ 4

*n* − (5 +

*n*) = (1 +

*kn*^{2}) − 4

*n*∴ 7

*n* –

*kn*^{2 }= 6 …(i)

Also, (1 +

*kn*^{2}) − 4

*n* = (

*n*^{2 }+2

*n* +2) − (1 +

* kn*^{2})

∴ 2

*kn*^{2 }=

*n*^{2} + 6

*n *

∴ 2

*k**n* =

*n* + 6 ...[∵

*n* ≠ 0]

∴ 2

*k**n* –

*n *= 6 ...(ii)

Equating (i) and (ii), we get

7

*n* –

*kn*^{2} = 2

*kn* –

*n*∴ 7 –

*kn* = 2

*k* –1 ...[∵

*n* ≠ 0]

∴ 2

*k* – 8 +

*kn* = 0

For

* k *to be an integer, we can take

*n* = 0, 2 or 6.

But

*n* = 0 is not possible.

For

*n* = 2,

* k *= 2 and for

*n* = 6,

*k* = 1

Since

*n* is the base of the number system and

*k*01 is the number expressed in the system to the base

*n*,

*k* <

*n*. Therefore,

*n = *2 is not possible.

So

*k* = 1 and

*n* = 6.

Therefore expressing the number of chocolates in the decimal system, we get

(15)

_{6 }= 11

(40)

_{6 }= 24

(

*k*01)

_{6 }=

_{ }(101)

_{6} = 37

(122)

_{6} = 50

So the number of chocolates with Jadoo, Badoo, Kadoo and Ladoo are 11, 24, 37 and 50, respectively.

∴The total number of chocolates that Aarav counted in the decimal number system = 11 + 24 + 37 + 50 = 122

∴ Option 1 can be eliminated.

Now (122)

_{10} can be expressed ass 442 and 233 in base 5 and base 7 respectively.

∴ Options 3 and 2 can also be eliminated.

Now (122)

_{10} = (322)

_{6 }= (233)

_{7}∴ (122)

_{10} cannot take any value between 233 and 322 in any base.

∴ 278 cannot be a valid representation of (122)

_{10 } Hence, option 4.