Of the 14 articles, 10 are of the same type and each of the remaining is of different type, find the number of combinations if 10 article are taken at a time.

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{I} \mathrm{guess} \mathrm{question} \mathrm{is} \mathrm{numbers} \mathrm{of} \mathrm{ways} \mathrm{to} \mathrm{select} 10 \mathrm{articles}. \\ \mathrm{Required} \mathrm{Answer} \\ =\left(4 \mathrm{different} \mathrm{type} \mathrm{and} 6 \mathrm{Other} \mathrm{type}\right) \mathrm{OR}\hspace{0.17em}\left(3 \mathrm{different} \mathrm{type} \mathrm{and} 7 \mathrm{Other} \mathrm{type}\right) \\ \mathrm{OR} \left(2 \mathrm{different} \mathrm{type} \mathrm{and} 8 \mathrm{Other} \mathrm{type}\right) \mathrm{OR} \left(1 \mathrm{different} \mathrm{type} \mathrm{and} 9 \mathrm{Other} \mathrm{type}\right) \\ \mathrm{OR} \left(10 \mathrm{same} \mathrm{type}\right) \\ \mathrm{Note} \mathrm{that} \mathrm{number} \mathrm{of} \mathrm{ways} \mathrm{to} \mathrm{select} \mathrm{any} \mathrm{number} \mathrm{of} \mathrm{articles} \mathrm{of} \mathrm{same} \mathrm{type} \mathrm{will} \mathrm{be} 1. \\ \Rightarrow \mathrm{Required} \mathrm{Answer} \\ {=}^4{\mathrm{C}}_4\times 1{+}^4{\mathrm{C}}_3\times 1{+}^4{\mathrm{C}}_2\times 1{+}^4{\mathrm{C}}_1\times 1+1 \\ {=}^4{\mathrm{C}}_4{+}^4{\mathrm{C}}_3{+}^4{\mathrm{C}}_2{+}^4{\mathrm{C}}_1+1 \\ =1+\frac{4.3.2}{3.2.1}+\frac{4.3}{2.1}+4+1 \\ =1+4+6+4+1 \\ =\mathbf{16}\mathbf{ }\left(\mathbf{Answer}\right) \end{gathered}$$
​

Hope this information will clear your doubts about this topic.

If you have any doubts just ask here on the ask and answer forum and our experts will try to help you out as soon as possible.
Regards