Question

Find the coefficient of \[{x^{100}}\] in the expression of \[\sum\limits_{j = 0}^{200} {{{\left( {1 + x} \right)}^j}} \].
A. \[{}^{200}{C_{100}}\]
B. \[{}^{201}{C_{102}}\]
C. \[{}^{200}{C_{101}}\]
D. \[{}^{201}{C_{100}}\]

Answer 1

Hint: First, use the binomial expansion formula for \[{\left( {1 + x} \right)^n}\] and expand the given expression. Check the pattern of the coefficients of each term of the expansion. Substitute the values of the total number of terms and value of the exponent of the required term in the formula of the coefficient and get the required answer.

Formula used:
 Binomial expansion of \[\sum\limits_{n = 0}^m {{{\left( {1 + x} \right)}^n}} \] with \[\left( {m + 1} \right)\] terms:
\[\sum\limits_{n = 0}^m {{{\left( {1 + x} \right)}^n}} = {}^m{C_0} + {}^m{C_1}x + {}^m{C_2}{x^2} + ... + {}^m{C_r}{x^r} + ... + {}^m{C_m}{x^m}\], where \[{}^m{C_r} = \dfrac{{m!}}{{r!\left( {m - r} \right)!}}\]

Complete step by step solution:
The given expression is \[\sum\limits_{j = 0}^{200} {{{\left( {1 + x} \right)}^j}} \].
Let’s apply the formula for the binomial expansion of \[\sum\limits_{n = 0}^m {{{\left( {1 + x} \right)}^n}} \].
Here, \[m = 200\]
We get,
\[\sum\limits_{j = 0}^{200} {{{\left( {1 + x} \right)}^j}} = {}^{200}{C_0} + {}^{200}{C_1}x + {}^{200}{C_2}{x^2} + ... + {}^{200}{C_r}{x^r} + ... + {}^{200}{C_{200}}{x^{200}}\]
From the above binomial expansion, we observe that
The coefficient of each term \[{x^r}\] is in the form \[{}^m{C_r}\].
Therefore, the coefficient of \[{x^{100}}\] in the expression \[\sum\limits_{j = 0}^{200} {{{\left( {1 + x} \right)}^j}} \]is:
The coefficient of \[{x^{100}} = {}^{200}{C_{100}}\]
Hence the correct option is A.

Additional Information:
Summation: The summation is a process of adding up a sequence of given numbers, the result is their total sum. The summation is represented by the symbol \[\Sigma \].
The summation notation is used to write the short form of the addition of the terms of a large sequence.

Combination: The combination is the method of determining the number of different ways of arranging and selecting objects out of a given number of objects, without considering the order of objects.
The number of combinations of \[n\] different objects taken \[r\] objects out of them without replacement, where \[0 < r \le n\], is denoted by: \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]

Note: We can also use the binomial expansion of \[{\left( {1 + x} \right)^n}\] in term of the summation.
Formula: \[{\left( {1 + x} \right)^n} = \sum\limits_{j = 0}^n {{}^n{C_j}{x^j}} \], where \[{}^n{C_j} = \dfrac{{n!}}{{j!\left( {n - j} \right)!}}\]