The number of distinct terms in the expansion of (X+Y+Z+W)

^{30}is:
(1)4060 (2)5456 (3)27405 (4)46376

Solution follows here:

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 x^{0}y^{n}
Considering
(X+Y) as first term and (Z+W) as second term observe the following binomial
expansions:

(X+Y+Z+W)

^{1}= 1C0 (X+Y)(Z+W)^{0}+ 1C1 (X+Y)^{0}(Z+W)
Number
of terms:
2*1
+ 1*2

(X+Y+Z+W)

^{2}= 2C0 (X+Y)^{2}(Z+W)^{0}^{ }+ 2C1 (X+Y)^{1}(Z+W)^{1}+ 2C2 (X+Y)^{0}(Z+W)^{2}
No.of
terms:
3*1
+
2*2 + 1*3

(X+Y+Z+W)

^{3}= 3C0 (X+Y)^{3}(Z+W)^{0 }+ 3C1 (X+Y)^{2}(Z+W) + 3C2 (X+Y)^{1}(Z+W)^{2}+ 3C3 (X+Y)^{0}(Z+W)^{3}
No.of
terms: 4*1 + 3*2
+ 2*3
+ 1*4

If
we observe this pattern, summary is:

(X+Y+Z+W)

**yields 2*1 + 1*2 terms**^{1}
(X+Y+Z+W)

**yields 3*1 + 2*2 + 1*3 terms**^{2}
(X+Y+Z+W)

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

(X+Y+Z+W)

**yields**^{30}
31*1
+ 30*2 + .....+ 17*15 + 16*16 +..........+ 1*31 terms

=
2{1*31 + 2*30 +....15*17} + 16*16

=
2{

_{1}∑^{17 }n(32-n)} + 16*16
=
2{(

_{1}∑^{17 }32*n) - (_{1}∑^{17 }n^{2})} + 256
=
2{32(

_{1}∑^{17 }n) - (_{1}∑^{17 }n^{2})} + 256 = 2{32(n(n+1)/2)_{n=17}- (n(n+1)(2n+1)/6)_{n=17}} +256
=
5456 terms

∴Distinct number of terms in (X+Y+Z+W)

^{30}= 5456**Answer (2)**
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