Question
Match each item given under the column $C_1$ to its correct answer given under the column $C_2.$
There are $10$ professors and $20$ lecturers out of whom a committee of $2$ professors and $3$ lecturer is to be formed. Find:
 
$C_1$
  $C_2$
$(a)$
In how many ways committee can be formed.
$(i)$ $^{10}C_2 \times ^{19}C_3$
$(b)$
In how many ways a particular professor is included.
$(ii)$ $^{10}C_2 \times ^{19}C_2$
$(c)$
In how many ways a particular lecturer is included.
$(iii)$ $^9C_1 \times ^{20}C_3$
$(d)$
In how many ways a particular lecturer is excluded.
$(iv)$ $^{10}C_2\times ^{20}C_3$

Answer

 
$C_1$
  $C_2$
$(a)$ In how many ways committee can be formed. $(iv)$ $^{10}C_2\times ^{20}C_3$
$(b)$ In how many ways a particular professor is included. $(iii)$ $^9C_1 \times ^{20}C_3$
$(c)$ In how many ways a particular lecturer is included. $(ii)$ $^{10}C_2 \times ^{19}C_2$
$(d)$ In how many ways a particular lecturer is excluded. $(i)$ $^{10}C_2 \times ^{19}C_3$
  1. We have to select $2$ professor out of $10$ and $3$ lecturers out of $20$
  2. $\therefore$ Number of ways of selection $=\ ^{10}C_2 \times ^{20}C_3$
  3. When a paeticular professor is included the number of ways $=\ ^{10 - 1}C_{1}\times ^{20}C_3 =\ ^9C_1 \times ^{20}C_3$
  4. When a particular lecturer is included number of ways $=\ ^{10}C_2 \times ^{19}C_2$
  5. Whan a particular lecturer is excluded, then number of ways $=\ ^{10}C_2 \times ^{19}C_3$

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