Number of ways of selecting $3$ balls out of $9$ balls without any restriction is ${\mathrm{}}^9{C}_3.$
Number of ways of selecting $3$ balls such that no black ball is selected is ${\mathrm{}}^6{C}_3.$

Number of ways such that atleast one black ball is selected is ${}^{9}C{}_3-{}^{6}C{}_3=84-20=64$