Permutation Combination Question 1
Question 1 - 2024 (01 Feb Shift 1)
If $n$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $n$ is equal to:
(1) 47
(2) 53
(3) 51
(4) 43
Show Answer
Answer (3)
Solution
Total ways to partition 5 into 4 parts are :
$5,0,0,0 \Rightarrow 1$ way
$4,1,0,0 \Rightarrow \frac{5 !}{4 !}=5$ ways
$3,2,0,0, \Rightarrow \frac{5 !}{3 ! 2 !}=10$ ways
$2,2,0,1 \Rightarrow \frac{5 !}{2 ! 2 ! 2 !}=15$ ways
$2,1,1,1 \Rightarrow \frac{5 !}{2 !(1 !)^{3} 3 !}=10$ ways
$3,1,1,0 \Rightarrow \frac{5 !}{3 ! 2 !}=10$ ways
Total $\Rightarrow 1+5+10+15+10+10=51$ ways
