Question

The value of $\frac{{18}^3+7^3+\mathrm{3.18.7.25}}{3^6+\mathrm{6.243.2}+\mathrm{15.81.4}+\mathrm{20.27.8}+\mathrm{15.9.16}+\mathrm{6.3.32}+64}$

• A. 4
• B. 3
• C. 2
• D. 1

Answer 1

Answer: (D)

1

note:- $(18+7{)}^3$ = ${18}^3+7^3+{3.18}^2.7+3.18.7^2$

$={18}^3+7^3+\mathrm{3.18.7}(18+7)$

$={18}^3+7^3+\mathrm{3.18.7.25.}$

Using binomial theorem,

$(3+2{)}^6$ = ${}^6{C}_0{3}^6$ + ${}^6{C}_1{3}^5{.2}^1{+}^6{C}_2{3}^4{.2}^2$ + ${}^6{C}_3{3}^5{.2}^3$ + ${}^6{C}_4{3}^2{2}^4$ + ${}^6{C}_5{3.2}^5$ + ${}^6{C}_6{2}^6.$

$=3^6+\mathrm{6.243.2}+\mathrm{15.81.4}+\mathrm{2..27.8}+\mathrm{15.9.16}+\mathrm{6.3.32}+64$

$\therefore \frac{{18}^3+7^3+\mathrm{3.18.7.25}}{3^6+\mathrm{6.243.2}+\mathrm{15.81.4}+\mathrm{20.27.8}+\mathrm{15.9.16}+\mathrm{6.3.32}+64.}$

$=\frac{(18+7{)}^3}{(3+2{)}^6}=\frac{(25{)}^3}{5^6}=\frac{(5^2{)}^3}{5^6}=\frac{5^6}{5^6}=1.$