Question

The total number of ways of dividing $15$ different things into groups of $8,4$ and $3$ respectively is

• A. ${}^{15}{C}_8\cdot {}^7{C}_4$
• B. ${}^{15}{C}_8\cdot {}^8{C}_4$
• C. ${}^{15}{C}_6\cdot {}^7{C}_4$
• D. ${}^{15}{C}_6\cdot {}^8{C}_4$

Answer 1

The correct option is A ${}^{15}{C}_8\cdot {}^7{C}_4$

Required number of ways
$=\frac{15!}{8!4!3!}=\frac{15!}{8!\cdot 7!}\times \frac{7!}{4!\cdot 3!}={}^{15}{C}_8\cdot {}^7{C}_4$

Alternate Solution:
Forming a group of $8$ object from $15$ can be done in ${}^{15}{C}_8$ ways and another group of $4$ objects can be formed from the rest of the objects left in ${}^7{C}_4$ ways, now last group of $3$ object can be formed in ${}^3{C}_3$ ways
Hence total possible ways are
$={}^{15}{C}_8\cdot {}^7{C}_4$