Question

$6$ Letters are to be posted in three boxes. The numbers of ways of posting the letters when no letterbox remains empty is

• A. 270
• B. 540
• C. 537
• D. None of these

Answer 1

The correct option is B 540

Since there are three letter box and six letters.

So, the possible cases without repetition are written below-

When the three letter box consists of $1,1,4$ letter respectively

Hence, number of ways $=\frac{6!}{4!\times 1!\times 1!\times 2!}\Rightarrow 15,$ $\left[Since1isrepeatedtwice\right]$

When the three letter box consists of $1,2,3$ letter respectively

Hence, number of ways $=\frac{6!}{1!\times 2!\times 3!}\Rightarrow 60.$

When the three letter box consists of $2,2,2$ letter respectively

Hence, number of ways $=\frac{6!}{2!\times 2!\times 2!\times 3!}\Rightarrow 15.$ $\left[Since2isrepeatedthrice\right]$

$\because$ All three letter boxes are different, .

$\therefore$ Number of ways of arrangement $=3!$

Hence,

Total number of ways of posting a letter such that no letter box remains empty $=(15+60+15)\times 3!\Rightarrow 540.$