Single Choice

The number of terms in the expansion off $$(1+x)^{101}(1+x^2-x)^{100}$$ in power of x is :

A$$302$$
B$$301$$
C$$202$$
Correct Answer
D$$101$$

Solution

$${ (1+x) }^{ 101 }{ (1-x+{ x }^{ 2 }) }^{ 100 }$$ can be written as

$$=(1+x){ (1+x) }^{ 100 }{ (1-x+{ x }^{ 2 }) }^{ 100 }$$

$$=(1+x)[(1+x){ (1-x+{ x }^{ 2 })] }^{ 100 }$$

$$=(1+x){ (1+{ x }^{ 3 }) }^{ 100 }$$

$$={ (1+{ x }^{ 3 }) }^{ 100 }+x{ (1+{ x }^{ 3 }) }^{ 100 }$$

Number of terms in $${ (1+{ x }^{ 3 }) }^{ 100 } = 101$$[terms are $$a_0+a_1x^3+a_2x^6+............+a_100x^{300}$$]

Number of terms in $$x{ (1+{ x }^{ 3 }) }^{ 100 } = 101$$[terms are $$a_0x+a_1x^4+a_2x^7+............+a_100x^{301}$$]

since there are no common terms
.
So, number of terms in $$(1+x){ (1+{ x }^{ 3 }) }^{ 100 }= 101 + 101 =202$$

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