nC2×nC(n-2) + 2×nC2×nC3 + nC3×nC(n-3)=100 30/09/2021 Bởi Amara nC2×nC(n-2) + 2×nC2×nC3 + nC3×nC(n-3)=100
Đáp án: Giải thích các bước giải: \[\begin{array}{l} C_n^2.C_n^{n – 2} + 2C_n^2.C_n^3 + C_n^3.C_n^{n – 3} = 100\\ \Leftrightarrow C_n^2.C_n^2 + 2C_n^2.C_n^3 + C_n^3.C_n^3 = 100\\ \Leftrightarrow {\left( {C_n^2} \right)^2} + 2C_n^2.C_n^3 + {\left( {C_n^3} \right)^2} = 100\\ \Leftrightarrow {\left( {C_n^2 + C_n^3} \right)^2} = 100\\ \Leftrightarrow C_n^2 + C_n^3 = 10\\ \Leftrightarrow \frac{{n!}}{{2!\left( {n – 2} \right)!}} + \frac{{n!}}{{3!\left( {n – 3} \right)!}} = 10\\ \Leftrightarrow \frac{{n\left( {n – 1} \right)}}{2} + \frac{{n\left( {n – 1} \right)\left( {n – 2} \right)}}{6} = 10\\ \Leftrightarrow 3n\left( {n – 1} \right) + n\left( {n – 1} \right)\left( {n – 2} \right) = 60\\ \Leftrightarrow 3{n^2} – 3n + {n^3} – 3{n^2} + 2n = 60\\ \Leftrightarrow {n^3} – n – 60 = 0\\ \Leftrightarrow n = 4 \end{array}\] Bình luận
Đáp án:
Giải thích các bước giải:
\[\begin{array}{l}
C_n^2.C_n^{n – 2} + 2C_n^2.C_n^3 + C_n^3.C_n^{n – 3} = 100\\
\Leftrightarrow C_n^2.C_n^2 + 2C_n^2.C_n^3 + C_n^3.C_n^3 = 100\\
\Leftrightarrow {\left( {C_n^2} \right)^2} + 2C_n^2.C_n^3 + {\left( {C_n^3} \right)^2} = 100\\
\Leftrightarrow {\left( {C_n^2 + C_n^3} \right)^2} = 100\\
\Leftrightarrow C_n^2 + C_n^3 = 10\\
\Leftrightarrow \frac{{n!}}{{2!\left( {n – 2} \right)!}} + \frac{{n!}}{{3!\left( {n – 3} \right)!}} = 10\\
\Leftrightarrow \frac{{n\left( {n – 1} \right)}}{2} + \frac{{n\left( {n – 1} \right)\left( {n – 2} \right)}}{6} = 10\\
\Leftrightarrow 3n\left( {n – 1} \right) + n\left( {n – 1} \right)\left( {n – 2} \right) = 60\\
\Leftrightarrow 3{n^2} – 3n + {n^3} – 3{n^2} + 2n = 60\\
\Leftrightarrow {n^3} – n – 60 = 0\\
\Leftrightarrow n = 4
\end{array}\]