nC2×nC(n-2) + 2×nC2×nC3 + nC3×nC(n-3)=100

Đáp án:

Giải thích các bước giải: \[\begin{array}{l} C_n^2.C_n^{n - 2} + 2C_n^2.C_n^3 + C_n^3.C_n^{n - 3} = 100\\ \Leftrightarrow C_n^2.C_n^2 + 2C_n^2.C_n^3 + C_n^3.C_n^3 = 100\\ \Leftrightarrow {\left( {C_n^2} \right)^2} + 2C_n^2.C_n^3 + {\left( {C_n^3} \right)^2} = 100\\ \Leftrightarrow {\left( {C_n^2 + C_n^3} \right)^2} = 100\\ \Leftrightarrow C_n^2 + C_n^3 = 10\\ \Leftrightarrow \frac{{n!}}{{2!\left( {n - 2} \right)!}} + \frac{{n!}}{{3!\left( {n - 3} \right)!}} = 10\\ \Leftrightarrow \frac{{n\left( {n - 1} \right)}}{2} + \frac{{n\left( {n - 1} \right)\left( {n - 2} \right)}}{6} = 10\\ \Leftrightarrow 3n\left( {n - 1} \right) + n\left( {n - 1} \right)\left( {n - 2} \right) = 60\\ \Leftrightarrow 3{n^2} - 3n + {n^3} - 3{n^2} + 2n = 60\\ \Leftrightarrow {n^3} - n - 60 = 0\\ \Leftrightarrow n = 4 \end{array}\]