119372 The total number of terms in the expansion of \((x+y)^{100}+(x-y)^{100}\) after simplification is

119373 Let \(a, b, c, d\) be real numbers such that
\(\sum_{\mathbf{k}=1}^{\mathbf{n}}\left(\mathbf{a} \mathbf{k}^3+\mathbf{b} \mathbf{k}^2+\mathbf{c k}+\mathbf{d}\right)=\mathbf{n}^4,\)
for every natural number \(n\). Then \(|\mathbf{a}|+|\mathbf{b}|+|\mathbf{c}|+|\mathbf{d}|\) is equal to

119374 The probability function of a binomial distribution is \(P(x)=\left(\begin{array}{l}6 \\ x\end{array}\right) p^x q^{6-x}\), \(x=0,1,2, \ldots, 6\). If \(2 P(2)=3 P(3)\), then \(p=\)

119375 Let \(\left(1+x+x^2\right)^n=a_0+a_1 x+a_2 x^2+\ldots+a_{2 n} x^{2 n}\). Then match the items of List-I with those of List-II
| | List-I | | List- |
| :---: | :---: | :---: | :---: |
| (A) | $a_0+a_2+\ldots+a_{2 n}$ | (I) | $\sqrt{n 3^{n-1}}$ |
| (B) | $a_1+a_3+\ldots+a_{2 n-1}$ | (II) | $\mathrm{n} 3^{\mathrm{n}}$ |
| (C) | $\mathrm{a}_1+2 \mathrm{a}_2+3 \mathrm{a}_3$ \lt br> $+\ldots+2 \mathrm{na}_{2 \mathrm{n}}$ | (III) | $\frac{1}{2}\left(3^{\mathrm{n}}+1\right)$ |
| | | (IV) | $\frac{1}{2}\left(3^n-1\right)$ |

119372 The total number of terms in the expansion of \((x+y)^{100}+(x-y)^{100}\) after simplification is

119373 Let \(a, b, c, d\) be real numbers such that
\(\sum_{\mathbf{k}=1}^{\mathbf{n}}\left(\mathbf{a} \mathbf{k}^3+\mathbf{b} \mathbf{k}^2+\mathbf{c k}+\mathbf{d}\right)=\mathbf{n}^4,\)
for every natural number \(n\). Then \(|\mathbf{a}|+|\mathbf{b}|+|\mathbf{c}|+|\mathbf{d}|\) is equal to

119374 The probability function of a binomial distribution is \(P(x)=\left(\begin{array}{l}6 \\ x\end{array}\right) p^x q^{6-x}\), \(x=0,1,2, \ldots, 6\). If \(2 P(2)=3 P(3)\), then \(p=\)

119375 Let \(\left(1+x+x^2\right)^n=a_0+a_1 x+a_2 x^2+\ldots+a_{2 n} x^{2 n}\). Then match the items of List-I with those of List-II
| | List-I | | List- |
| :---: | :---: | :---: | :---: |
| (A) | $a_0+a_2+\ldots+a_{2 n}$ | (I) | $\sqrt{n 3^{n-1}}$ |
| (B) | $a_1+a_3+\ldots+a_{2 n-1}$ | (II) | $\mathrm{n} 3^{\mathrm{n}}$ |
| (C) | $\mathrm{a}_1+2 \mathrm{a}_2+3 \mathrm{a}_3$ \lt br> $+\ldots+2 \mathrm{na}_{2 \mathrm{n}}$ | (III) | $\frac{1}{2}\left(3^{\mathrm{n}}+1\right)$ |
| | | (IV) | $\frac{1}{2}\left(3^n-1\right)$ |

119372 The total number of terms in the expansion of \((x+y)^{100}+(x-y)^{100}\) after simplification is

119373 Let \(a, b, c, d\) be real numbers such that
\(\sum_{\mathbf{k}=1}^{\mathbf{n}}\left(\mathbf{a} \mathbf{k}^3+\mathbf{b} \mathbf{k}^2+\mathbf{c k}+\mathbf{d}\right)=\mathbf{n}^4,\)
for every natural number \(n\). Then \(|\mathbf{a}|+|\mathbf{b}|+|\mathbf{c}|+|\mathbf{d}|\) is equal to

119374 The probability function of a binomial distribution is \(P(x)=\left(\begin{array}{l}6 \\ x\end{array}\right) p^x q^{6-x}\), \(x=0,1,2, \ldots, 6\). If \(2 P(2)=3 P(3)\), then \(p=\)

119375 Let \(\left(1+x+x^2\right)^n=a_0+a_1 x+a_2 x^2+\ldots+a_{2 n} x^{2 n}\). Then match the items of List-I with those of List-II
| | List-I | | List- |
| :---: | :---: | :---: | :---: |
| (A) | $a_0+a_2+\ldots+a_{2 n}$ | (I) | $\sqrt{n 3^{n-1}}$ |
| (B) | $a_1+a_3+\ldots+a_{2 n-1}$ | (II) | $\mathrm{n} 3^{\mathrm{n}}$ |
| (C) | $\mathrm{a}_1+2 \mathrm{a}_2+3 \mathrm{a}_3$ \lt br> $+\ldots+2 \mathrm{na}_{2 \mathrm{n}}$ | (III) | $\frac{1}{2}\left(3^{\mathrm{n}}+1\right)$ |
| | | (IV) | $\frac{1}{2}\left(3^n-1\right)$ |

119372 The total number of terms in the expansion of \((x+y)^{100}+(x-y)^{100}\) after simplification is

119373 Let \(a, b, c, d\) be real numbers such that
\(\sum_{\mathbf{k}=1}^{\mathbf{n}}\left(\mathbf{a} \mathbf{k}^3+\mathbf{b} \mathbf{k}^2+\mathbf{c k}+\mathbf{d}\right)=\mathbf{n}^4,\)
for every natural number \(n\). Then \(|\mathbf{a}|+|\mathbf{b}|+|\mathbf{c}|+|\mathbf{d}|\) is equal to

119374 The probability function of a binomial distribution is \(P(x)=\left(\begin{array}{l}6 \\ x\end{array}\right) p^x q^{6-x}\), \(x=0,1,2, \ldots, 6\). If \(2 P(2)=3 P(3)\), then \(p=\)

119375 Let \(\left(1+x+x^2\right)^n=a_0+a_1 x+a_2 x^2+\ldots+a_{2 n} x^{2 n}\). Then match the items of List-I with those of List-II
| | List-I | | List- |
| :---: | :---: | :---: | :---: |
| (A) | $a_0+a_2+\ldots+a_{2 n}$ | (I) | $\sqrt{n 3^{n-1}}$ |
| (B) | $a_1+a_3+\ldots+a_{2 n-1}$ | (II) | $\mathrm{n} 3^{\mathrm{n}}$ |
| (C) | $\mathrm{a}_1+2 \mathrm{a}_2+3 \mathrm{a}_3$ \lt br> $+\ldots+2 \mathrm{na}_{2 \mathrm{n}}$ | (III) | $\frac{1}{2}\left(3^{\mathrm{n}}+1\right)$ |
| | | (IV) | $\frac{1}{2}\left(3^n-1\right)$ |