Question

The number of students who joined the 'KEEP YOUR PLACE CLEAN' campaign are represented by 13th term in the expansion of ${(9x-\frac{1}{3\sqrt{x}})}^{18}$, Find the number of students.

Answer 1

The 13th term in the expansion ${(9x-\frac{1}{3\sqrt{x}})}^{18}$ is

$T_{13}=T_{12+1}{=}^{18}{C}_{12}(9x{)}^{18-12}\times {\left(\frac{-1}{2\sqrt{x}}\right)}^{12}$

${=}^{18}{C}_{12}.{\frac{9^6\times x^6\times (-1{)}^{12}}{3^{12}\times (\sqrt{x}}}^{12}$

${=}^{18}{C}_{12}\times \frac{(3^2{)}^6}{3^{12}}{=}^{18}{C}_{12}{=}^{18}{C}_6\left[{\because }^n{C}_r{=}^n{C}_{n-r}\right]$

=$\frac{18\times 17\times 16\times 15\times 14\times 13}{6\times 5\times 4\times 6\times 3\times 2\times 1}=18564$