Single Choice

If $$n$$ is an even integer and $$a,b,c$$ are distinct, the number of distinct terms in the expansion of $$(a+b+c)^n+(a+b-c)^n$$ is

A$$\displaystyle { \left( \frac { n }{ 2 } \right) }^{ 2 }$$
B$$\displaystyle { \left( \frac { n+1 }{ 2 } \right) }^{ 2 }$$
C$$\displaystyle { \left( \frac { n+2 }{ 2 } \right) }^{ 2 }$$
Correct Answer
D$$\displaystyle { \left( \frac { n+3 }{ 2 } \right) }^{ 2 }$$

Solution

Let $$n=2m$$ $$m\in N$$
Therefore $${ \left( a+b+c \right) }^{ n }+{ \left( a+b-c \right) }^{ n }={ \left( a+b+c \right) }^{ 2m }+{ \left( a+b-c \right) }^{ 2m }$$
$$=2\left[ { \left( a+b \right) }^{ 2m }+_{ }^{ 2m }{ { C }_{ 2 }^{ } }{ \left( a+b \right) }^{ 2m-2 }.{ c }^{ 2 }+...+^{ 2m }{ { C }_{ 2m }^{ } }.{ c }^{ 2m } \right] $$
Therefore the number of distinct terms
$$\left( 2m+1 \right) +\left( 2m-1 \right) +..3+2+1=\displaystyle\frac { m+1 }{ 2 } \left( 2m+2 \right) ={ \left( m+1 \right) }^{ 2 }$$
Susbtitute in terms of n to get $$={ \left( \displaystyle\frac { n }{ 2 } +1 \right) }^{ 2 }={ \left( \displaystyle\frac { n+2 }{ 2 } \right) }^{ 2 }$$

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