Question:
How many four digit numbers $abcd$ exist such that $a$ is odd, $b$ is divisible by $3$, $c$ is even and $d$ is prime?
How many four digit numbers $abcd$ exist such that $a$ is odd, $b$ is divisible by $3$, $c$ is even and $d$ is prime?
Updated On: Jun 7, 2024
- 380
- 360
- 400
- 520
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The Correct Option is C
Solution and Explanation
We know that, the odd numbers are
$ \{1,3,5,7,9\} $
$\therefore n(a)=5$
Divisible by $3$ are $\{0,3,6,9\}, n(b)=4$
Even numbers are $\{0,2,4,6,8\}, n(c)=5$
and prime numbers are $\{2,3,5,7\}, n(d)=4$
$\therefore$ Four-digit numbers $abcd$ exist
$=5 \cdot 4 \cdot 5 \cdot 4$
$=400$
$ \{1,3,5,7,9\} $
$\therefore n(a)=5$
Divisible by $3$ are $\{0,3,6,9\}, n(b)=4$
Even numbers are $\{0,2,4,6,8\}, n(c)=5$
and prime numbers are $\{2,3,5,7\}, n(d)=4$
$\therefore$ Four-digit numbers $abcd$ exist
$=5 \cdot 4 \cdot 5 \cdot 4$
$=400$
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Concepts Used:
Permutations and Combinations
Permutation:
Permutation is the method or the act of arranging members of a set into an order or a sequence.
- In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point.
- A permutation is used in many events of daily life. It is used for a list of data where the data order matters.
Combination:
Combination is the method of forming subsets by selecting data from a larger set in a way that the selection order does not matter.
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- The combination is used for a group of data where the order of data does not matter.