Question

To make a 750 - piece jigsaw puzzle more challenging, a puzzle company includes 5 extra pieces of box along with the 750 pieces, and those extra pieces do not fit anywhere in the puzzle. If Ramesh selects one piece at random from such a box, what is the probability of selecting the extra piece?
A. $\dfrac{1}{5}$
B. $\dfrac{1}{755}$
C. $\dfrac{1}{750}$
D.$\dfrac{5}{755}$
E. $\dfrac{5}{750}$

Answer 1

Assume the number of favourable outcomes as $n\left( E \right)$ and the total number of outcomes as $n\left( S \right)$. Find the value of $n\left( E \right)$ by using the formula of combinations used for selecting 1 piece from the 5 extra pieces. Now, to calculate $n\left( S \right)$, first take the sum of 750 pieces with these 5 extra pieces and from a total of 755 pieces, use the formula for selecting 1 piece. Find the ratio of $n\left( E \right)$ to $n\left( S \right)$ to determine the required probability.

Complete step by step answer:
Here, we have been given a jigsaw puzzle having 750 pieces. 5 extra pieces were provided with these 750 pieces. These 5 extra pieces do not fit anywhere in the puzzle. We have to find the probability of selecting the extra piece by Ramesh. Now, Ramesh must select 1 piece from these 5 extra pieces. Therefore, we have,
Number of ways to select 1 piece from the 5 extra pieces = ${}^{5}{{C}_{1}}=5$.
This is the number of favourable outcomes of the given situation. Let us denote this number with $n\left( E \right)$.
$\Rightarrow n\left( E \right)=5$
Now, when these 5 extra pieces are added with 750 pieces, then the total number of pieces will be 755. So, the total number of outcomes, denoted by, $n\left( S \right)$, will be the total number of ways to select 1 piece from 755 pieces. So, we have,
Number of ways to select 1 piece from the 755 pieces = ${}^{755}{{C}_{1}}=755$.
This is the total number of outcomes denoted with $n\left( S \right)$.
$\Rightarrow n\left( S \right)=755$
We know that the probability of an event to occur is the ratio of the number of favourable outcomes to the total number of outcomes.
$\therefore \text{Probability}=\dfrac{n\left( E \right)}{n\left( S \right)}=\dfrac{5}{755}$
Hence, option D is the correct answer.

Note:
One may note that while calculating the total number of outcomes we have to consider the total number of pieces equal to 755 and not 750 because 5 pieces were added in extra. You must remember the formula of combinations so that the number of favourable outcomes and total number of outcomes can be calculated. Here, you must not apply the formula for permutation because we do not have to arrange but we have to select. So, formula for combinations is applied.