mathematics-class-11-unit-09-chapter-07-binomial-theorem-l-9-9-hxb8v1szx18

### <Success>Example 1</Success>
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<p>Simplify:
$2^k~ { }^n C_0 { }^n C_k - 2^{k-1} ~{ }^n C_1 { }^{n-1} C_{k-1} + 2^{k-1} { }^n C_2 { }^{n-2} C_k  … (-1)^k { }^n C_k { }^{n-k} C_0$</p>
<p><strong>Solution:</strong></p>
<p>$i^{th}$ term $\rightarrow \sum_{i=0}^{i-k} (-1)^i 2^{k-i} { }^n C_i { }^{n-i}C_{k-i}$</p>
<p>$\frac{k !}{k !}$ $\cdot \frac{n !}{i ! (n-i)!}$ $\cdot \frac{(n-i)!}{(k-i)!(n-k)}$</p>
<p>$\sum_{i=0}^k (-1)^i 2^{k-i} ~ { }^n C_k { }^k C_i$</p>
<p>$={ }^n C_k [ \sum_{i=0}^{k} (-1)^i 2^{k-i} { }^kC_i] = { }^n C_k$</p>
<p>$ { }^k C_0 \cdot 2^k (-1)^0 + { }^k C_1 2^{k-1} (-1)^1 + { }^k C_2 2^{k-2} (-1)^2 + …[2+(-1)]^k$</p>
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<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-09-chapter-07-binomial-theorem-L-9-9-hxb8v1szx18-3a.jpg">

### <Success>Solution</Success>
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<p>$ S_k =3^k { }^{100} C_0 { }^{100} C_k - 3^{k-1} { }^{100} C_1 { }^{99} C_{k-1} +…(-1)^k { }^{100} C_k { }^{100-k} C_0$</p>
<p>$= \sum_{i=0}^k 3^{k-i} \cdot (-1)^i \cdot { }^{100} C_i \cdot { }^{100-i} C_{k-i}$</p>
<p>$\frac{k!}{k!} \cdot \frac{100!}{i!(100-i)!} \cdot \frac{(100-i)!}{(k-i)!(100-k)!}$</p>
<p>$=\sum_{i=0}^k 3^{k-i} (-1)^i { }^{100} C_k \cdot { }^k C_i = { }^{100} C_k \cdot \sum_{i=0}^k 3^{k-i} \cdot (-1)^i { }^k C_i$</p>
<p>$(3-1)(3-1)….(3-1)$</p>
<p>$(3-1)(3-1)….(3-1) \rightarrow k$</p>
<p>$(3-1)(3-1).. \rightarrow k-i$</p>
<p>$…(3-1)\rightarrow i$</p>
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<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-09-chapter-07-binomial-theorem-L-9-9-hxb8v1szx18-5.jpg"></p>
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<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-09-chapter-07-binomial-theorem-L-9-9-hxb8v1szx18-5a.jpg"></p>

### <Success>Solution</Success>
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<p>${ }^{100} C_k \cdot \sum_{i=0}^k 3^{k-i} \cdot (-1)^i { }^k C_i \rightarrow (3-1)^k$<br>
$S_k={ }^{100} C_k \cdot 2^k$</p>
<p>$S_{100-k}={ }^{100} C_{100-k} \cdot 2^{100-k}$</p>
<p>$\sum_{k=0}^{100} S_k S_{100-k}=\sum_{k=0}^{100}{ }^{100} C_k \cdot{ }^{100} C_{100-k} \cdot 2^{100}$</p>
<p>$=2^{100} \sum_{k=0}^{100}{ }^{100} c_k{ }^{100} C_{100-k}$</p>
<p>$(x+1 / x)^{200}=(x+1 / x)^{100}(x+1 / x)^{100}$</p>
<p>${ }^{200} C_{100}=C_0 \cdot c_{100}+c_1 \cdot c_{99}+c_2 \cdot c_{98}+\cdot \cdot$</p>
<p>$={ }^{200} C_{100} \cdot 2^{100}$</p>




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<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-09-chapter-07-binomial-theorem-L-9-9-hxb8v1szx18-6.jpg"></p>
<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-09-chapter-07-binomial-theorem-L-9-9-hxb8v1szx18-6a.jpg"></p>

### <Success>Example 3</Success>
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<p>$ 1 + x, 1 + x + x^2, 1 + x + x^2 + x^3, …. 1 + x + x^2 + … x^n $</p>
<p>$ (1 + x)(1 + x + x^2)(1 + x + x^2 + x^3)(…) … (1 + x + x^2 + … + x^n)$</p>
<p>$= a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ….. a_N x^N$</p>
<ol>
<li>
<p>How many terms in the expansion?</p>
</li>
<li>
<p>Show that coeffs equi-distant are equal from beginning and end.</p>
</li>
<li>
<p>Sum of odd coeffs = Sum of even coeffs</p>
</li>
</ol>
<p>$=\frac{(n+1)!}{2}$</p>
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<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-09-chapter-07-binomial-theorem-L-9-9-hxb8v1szx18-8a.jpg"></p>

### <Success>Solution</Success>
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<p>$x^{1+2+3……n}= x^{\frac{n(n+1)}{2}}$</p>
<p>${\frac{n(n+1)}{2}}+1$</p>
<p>$(1+x)(1+x+x^2)(…)…(1+x+x^2 + x+…x^n=a_0 +a_1x + … a_N x^N$</p>
<p>$(1+x^{-1})(1+x^{-1} + x^{-2})…(1+x^{-1} +x^{-2} + … x^{-n})=a_0 + a_1 x^{-1}+…a_Nx^{-N}$</p>
<p>$x^{-N}(1+x)(1+x+x^2)…(x^n + x^{n-1} + … +x^2 +x +1)=x^{-N}(a_0 + a_1x + … a_N x^N)$</p>
<p>$0=a_0-a_1+a_2-a_3+… \Rightarrow$ Sum of odd terms</p>
<p>= Sum of even terms</p>
<p>$2 \times 3 \times 4 \times … (n+1) = 2 \times$ Sum of odd</p>
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<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-unit-09-chapter-07-binomial-theorem-L-9-9-hxb8v1szx18-9.jpg"></p>
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<p>======</p>
<h3 id="successthank-yousuccess"><Success>Thank you</Success></h3>
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