CAT 2025 Lesson : Permutations & Combinations - Blanks with Repetition

In how many ways can $6$ coins of different denominations be distributed to $4$ students?
($1$) $4^6$            ($2$) $6^4$            ($3$) $^{6} \text{P}_{4}$            ($4$) $^{6} \text{C}_{4}$           

Solution

Each coin can be assigned to $4$ different students. There are $6$ such coins.
$\therefore$ Total Permutations $= \underline{4} \times \underline{4} \times \underline{4} \times \underline{4} \times \underline{4} \times \underline{4} = 4^{6}$

Answer: ($1$) $4^6$

What is the number of ways in which a student can answer the Quant section of the CAT paper, which comprises of $34$ questions with $4$ options each, wherein she attempts one or more of the questions?

($1$) $4^{34}$            ($2$) $5^{34}$            ($3$) $4^{34} - 1$            ($4$) $5^{34} - 1$           

Solution

The total number of ways in which a student can answer a question is $5$ ways (Mark Option $1$, Option $2$, Option $3$, Option $4$ or leave it unattempted). As there are $34$ questions, the total number of ways she can answer the questions is $\underline{5} \times \underline{5} \times \underline{5} \times ... \ _{34 \space times}$ $= 5^{34}$

The question, however, requires us to find the cases where she attempts one or more questions. Therefore, we need to remove the $1$ case where she leaves all the questions unattempted.

$\therefore$ Total Permutations = $5^{34} – 1$

Answer: ($4$) $5^{34} - 1$