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

Solution follows here:

^{20}?
(1)231 (2)253 (3)242 (4)210 (5) 228

__Solution:__

This
is application of binomial theorem:

(x+y)

^{n}= nC0 x^{n}y^{0}+ nc1 x^{n-1}y +nC2 x^{n-2}y^{2}+….+ nCn y^{n}
Considering
(a+b) as first term and ‘c’ as second term observe the following binomial
expansions:

(a+b+c)

^{1}= 1C0 (a+b)^{1}c^{o}+ 1C1 (a+b)^{0}c^{1}
Number
of terms:
2*1
+ 1*1

(a+b+c)

^{2}= 2C0 (a+b)^{2}^{ }c^{o}+ 2C1 (a+b)^{1}c^{1}+ 2C2 (a+b)^{0}c^{2}
= 2C0 (a

^{2}+2ab+b^{2})*1 + 2C1 (a+b)*c + 2C2 (1*c^{2})
No. of terms: 3*1 + 2*1
+ 1*1

(a+b+c)

^{3}= 3C0 (a+b)^{3}c^{o}^{ }+ 3C1 (a+b)^{2}c + 3C2 (a+b)^{1}c^{2}+ 3C3 (a+b)^{0}c^{3}
= 3C0
(a

^{3}+b^{3}+3a^{2}b+3ab^{2}) c^{o}^{ }+ 3C1 (a^{2}+b^{2}+2ab)c + 3C2 (a+b)c^{2}+ 3C3 1* c^{3}
No.of terms: 4*1 +
3*1
+ 2*1 + 1*1

If
we observe this pattern, summary is:

(a+b+c)

**yields 2*1 + 1*1 = 2+1 terms**^{1}
(a+b+c)

**yields 3*1 + 2*1 + 1*1 = 3+2+1 terms**^{2}
(a+b+c)

**yields 4*1 + 3*1 + 2*1 + 1*1 = 4+3+2+1 terms**^{3}
Going
by this logic,

(a+b+c)

**yields (n+1)+n+(n-1)+……..+3+2+1 = ∑(n+1) terms**^{n}
=
(n+2)(n+1)/2
(as we know that Σn = (n+1)n/2)

Distinct
number of terms in (a+b+c)

^{20}= (20+2)(20+1)/2 = 11*21 = 231**Answer (1)**

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