119345 Let \(\mathbf{f}(\mathbf{x})=(1-\mathbf{x})^{\mathrm{n}}\), where \(\mathbf{n}\) is a non-negative integer. What is
\(\mathbf{f}(\mathbf{0})+\mathbf{f}^{\prime}(\mathbf{0})+\frac{\mathbf{f}^{\prime \prime}(\mathbf{0})}{2 !}+\ldots+\frac{\mathbf{f}^{\mathrm{n}}(\mathbf{0})}{\mathrm{n} !} \text { equal to? }\)

119346 If \(5^{99}\) is divided by 13 , then the remainder is

119347 \(\mathrm{C}_0, \mathrm{C}_1, \mathrm{C}_2 \ldots \ldots, \mathrm{C}_{15}\) are the binomial coefficients in the expansion of \((1+x)^{15}\), then \(\frac{\mathbf{C}_1}{\mathbf{C}_0}+\frac{{ }^2 \mathbf{C}_2}{\mathbf{C}_1}+\frac{{ }^3 \mathbf{C}_3}{\mathbf{C}_2}+\ldots .+\frac{{ }^{15} \mathbf{C}_{15}}{\mathbf{C}_{14}}\) is

119348 If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of \(\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^n\) is \(\sqrt{6}: 1\), then the third term from the beginning is :

119345 Let \(\mathbf{f}(\mathbf{x})=(1-\mathbf{x})^{\mathrm{n}}\), where \(\mathbf{n}\) is a non-negative integer. What is
\(\mathbf{f}(\mathbf{0})+\mathbf{f}^{\prime}(\mathbf{0})+\frac{\mathbf{f}^{\prime \prime}(\mathbf{0})}{2 !}+\ldots+\frac{\mathbf{f}^{\mathrm{n}}(\mathbf{0})}{\mathrm{n} !} \text { equal to? }\)

119346 If \(5^{99}\) is divided by 13 , then the remainder is

119347 \(\mathrm{C}_0, \mathrm{C}_1, \mathrm{C}_2 \ldots \ldots, \mathrm{C}_{15}\) are the binomial coefficients in the expansion of \((1+x)^{15}\), then \(\frac{\mathbf{C}_1}{\mathbf{C}_0}+\frac{{ }^2 \mathbf{C}_2}{\mathbf{C}_1}+\frac{{ }^3 \mathbf{C}_3}{\mathbf{C}_2}+\ldots .+\frac{{ }^{15} \mathbf{C}_{15}}{\mathbf{C}_{14}}\) is

119348 If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of \(\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^n\) is \(\sqrt{6}: 1\), then the third term from the beginning is :

119345 Let \(\mathbf{f}(\mathbf{x})=(1-\mathbf{x})^{\mathrm{n}}\), where \(\mathbf{n}\) is a non-negative integer. What is
\(\mathbf{f}(\mathbf{0})+\mathbf{f}^{\prime}(\mathbf{0})+\frac{\mathbf{f}^{\prime \prime}(\mathbf{0})}{2 !}+\ldots+\frac{\mathbf{f}^{\mathrm{n}}(\mathbf{0})}{\mathrm{n} !} \text { equal to? }\)

119346 If \(5^{99}\) is divided by 13 , then the remainder is

119347 \(\mathrm{C}_0, \mathrm{C}_1, \mathrm{C}_2 \ldots \ldots, \mathrm{C}_{15}\) are the binomial coefficients in the expansion of \((1+x)^{15}\), then \(\frac{\mathbf{C}_1}{\mathbf{C}_0}+\frac{{ }^2 \mathbf{C}_2}{\mathbf{C}_1}+\frac{{ }^3 \mathbf{C}_3}{\mathbf{C}_2}+\ldots .+\frac{{ }^{15} \mathbf{C}_{15}}{\mathbf{C}_{14}}\) is

119348 If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of \(\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^n\) is \(\sqrt{6}: 1\), then the third term from the beginning is :

119345 Let \(\mathbf{f}(\mathbf{x})=(1-\mathbf{x})^{\mathrm{n}}\), where \(\mathbf{n}\) is a non-negative integer. What is
\(\mathbf{f}(\mathbf{0})+\mathbf{f}^{\prime}(\mathbf{0})+\frac{\mathbf{f}^{\prime \prime}(\mathbf{0})}{2 !}+\ldots+\frac{\mathbf{f}^{\mathrm{n}}(\mathbf{0})}{\mathrm{n} !} \text { equal to? }\)

119346 If \(5^{99}\) is divided by 13 , then the remainder is

119347 \(\mathrm{C}_0, \mathrm{C}_1, \mathrm{C}_2 \ldots \ldots, \mathrm{C}_{15}\) are the binomial coefficients in the expansion of \((1+x)^{15}\), then \(\frac{\mathbf{C}_1}{\mathbf{C}_0}+\frac{{ }^2 \mathbf{C}_2}{\mathbf{C}_1}+\frac{{ }^3 \mathbf{C}_3}{\mathbf{C}_2}+\ldots .+\frac{{ }^{15} \mathbf{C}_{15}}{\mathbf{C}_{14}}\) is

119348 If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of \(\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^n\) is \(\sqrt{6}: 1\), then the third term from the beginning is :