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CAT 2008 — QA Question 15

Fundamental Principles of P&CEasy
Passage / Data

Directions for next 2 questions:

The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

What is the number of distinct terms in the expansion of (a + b + c)20?

Answer & solution

  • 231

  • B

    253

  • C

    242

  • D

    210

  • E

    228

Solution

Consider (a + b + c)20

Each term in the expansion of (a + b + c)20 can be written as 20Cr × ax × by × cz, where (x + y + z) = 20

Number of distinct terms in the expansion will be same as the number of distinct pairs of x, y and z.

∴ We have to divide 20 into three parts which can be done by using the distribution rule = n+r-1Cr-1

Where,

n is number of things to be distributed.

r is number of parts into which the things are to be distributed.

∴ To divide 20 into 3 parts we have,

20+3-1C3-1 = 22C2 = 22×212×1 = 11 × 21 = 231

Alternatively,
This can be solved without using much knowledge of permutations and combinations as follows,

(a + b + c)1 = a + b + c [i.e. 3 terms = (1 + 2) terms]

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac [i.e. 6 terms = (1 + 2 + 3) terms]

(a + b + c)3 = a3 + b3 + c3 + 6abc + 3ab2 + 3ac2 + 3a2b + 3bc2 + 3a2c + 3b2c [i.e. 10 terms = (1 + 2 + 3 + 4) terms]

Similarly,

(a + b + c)n will have (1 + 2 + 3 + … + (n + 1)) terms

∴ (a + b + c)20 will have (1 + 2 + 3 + … + 21) = 231 terms

Hence, option (a).

CAT 2008 QA Q15: What is the number of distinct terms in the expansion of (a + b + c) 20 ? — Solution | TheCATExam