数列$\left\{ {{a_n}} \right\}$共有$11$项，${a_1} = 0$，${a_{11}} = 4$，且$\left| {{a_{k + 1}} - {a_k}} \right| = 1$，$k = 1,2,\cdots,10$，满足这种条件的不同数列的个数为______．
正确答案是$120$．
分析与解 因为 ${a_{k + 1}} - {a_k} = 1$或$ - 1$，假设共有$x$个$1$，则$$x + \left( { - 1} \right)\left( {10 - x} \right) = 4,$$得$x = 7$．所以${\mathrm {C}}_{10}^7 = 120$为所求．