The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct is 

No of identical things = 10

No of distinct things = 21

No of ways of chosing 10 objects

= (10 identical, 0 different)+ (9 identical, 1 different)+ ….. +  (01 identical, 10 different)

$\begin{array}{rcl} & =& 1\left({21}_{{\mathrm{C}}_0}\right)+1\left({21}_{{\mathrm{C}}_1}\right)+1\left({21}_{{\mathrm{C}}_2}\right)+ ..... +1\left({21}_{{\mathrm{C}}_{10}}\right)\\ & =& \frac{1}{2}{(}^{21}{\mathrm{C}}_0+{21}_{{\mathrm{C}}_1}+{21}_{{\mathrm{C}}_2} .... {21}_{{\mathrm{C}}_{10}}+{21}_{{\mathrm{C}}_{11}}+ .... +{21}_{{\mathrm{C}}_{21}})\\ & =& \frac{1}{2}\left(2^{21}\right)=2^{20}\end{array}$