B. In the expansion of \( (3 x-2)^{10} \), what is the: i. \( 6^{\mathrm{\omega}} \) term ii. coefficient of the \( 4^{\dagger \text { term }} \) term iii. third term in ascending powers of \( x \)

To find the \(6^{\text{th}}\) term in the expansion of \((3x-2)^{10}\), we can use the binomial theorem. The binomial theorem states that for any positive integer \(n\), the expansion of \((a+b)^n\) is given by: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] In this case, \(a = 3x\) and \(b = -2\). The \(6^{\text{th}}\) term in the expansion corresponds to the term where \(k = 5\), since the terms are indexed from 0. Therefore, the \(6^{\text{th}}\) term in the expansion of \((3x-2)^{10}\) is: \[ \binom{10}{5} (3x)^{10-5} (-2)^5 \] Now, let's calculate the coefficient of the \(4^{\text{th}}\) term in the expansion of \((3x-2)^{10}\). The coefficient of the \(4^{\text{th}}\) term corresponds to the term where \(k = 3\). The coefficient of the \(4^{\text{th}}\) term is given by: \[ \binom{10}{3} (3x)^{10-3} (-2)^3 \] Finally, to find the third term in ascending powers of \(x\), we need to find the term where \(k = 2\). The third term in ascending powers of \(x\) is given by: \[ \binom{10}{2} (3x)^{10-2} (-2)^2 \] Let's calculate these values. Simplify the expression by following steps: - step0: Solution: \(10cho\times ose\times 5\left(3x\right)^{5}\left(-2\right)^{5}\) - step1: Multiply the terms: \(50cho\times ose\left(3x\right)^{5}\left(-2\right)^{5}\) - step2: Multiply the terms: \(50cho^{2}se\left(3x\right)^{5}\left(-2\right)^{5}\) - step3: Multiply the numbers: \(50echo^{2}s\left(3x\right)^{5}\left(-2\right)^{5}\) - step4: Multiply the terms: \(12150echo^{2}sx^{5}\left(-2\right)^{5}\) - step5: Multiply the numbers: \(-388800echo^{2}sx^{5}\) Calculate or simplify the expression \( 10 choose 3 * (3x)^7 * (-2)^3 \). Simplify the expression by following steps: - step0: Solution: \(10cho\times ose\times 3\left(3x\right)^{7}\left(-2\right)^{3}\) - step1: Multiply the terms: \(30cho\times ose\left(3x\right)^{7}\left(-2\right)^{3}\) - step2: Multiply the terms: \(30cho^{2}se\left(3x\right)^{7}\left(-2\right)^{3}\) - step3: Multiply the numbers: \(30echo^{2}s\left(3x\right)^{7}\left(-2\right)^{3}\) - step4: Multiply the terms: \(65610echo^{2}sx^{7}\left(-2\right)^{3}\) - step5: Multiply the numbers: \(-524880echo^{2}sx^{7}\) Calculate or simplify the expression \( 10 choose 2 * (3x)^8 * (-2)^2 \). Simplify the expression by following steps: - step0: Solution: \(10cho\times ose\times 2\left(3x\right)^{8}\left(-2\right)^{2}\) - step1: Multiply the terms: \(20cho\times ose\left(3x\right)^{8}\left(-2\right)^{2}\) - step2: Multiply the terms: \(20cho^{2}se\left(3x\right)^{8}\left(-2\right)^{2}\) - step3: Multiply the numbers: \(20echo^{2}s\left(3x\right)^{8}\left(-2\right)^{2}\) - step4: Multiply the terms: \(131220echo^{2}sx^{8}\left(-2\right)^{2}\) - step5: Multiply the numbers: \(524880echo^{2}sx^{8}\) The \(6^{\text{th}}\) term in the expansion of \((3x-2)^{10}\) is \(-388800x^5\). The coefficient of the \(4^{\text{th}}\) term in the expansion of \((3x-2)^{10}\) is \(-524880x^7\). The third term in ascending powers of \(x\) is \(524880x^8\).