Let S denote the set of 4-digit numbers abcd | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) For $n(S)$, we need to choose $4$ distinct digits from the set ${0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$. Since the condition $a > b > c > d$ is given, onc

Vedclass · Accepted Answer

$260$

Vedclass · Answer

(B) For $n(S)$, we need to choose $4$ distinct digits from the set ${0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$. Since the condition $a > b > c > d$ is given, once $4$ digits are chosen, there is only $1$ way to arrange them in descending order. Thus, $n(S) = {}^{10}C_4 = \frac{10 	imes 9 	imes 8 	imes 7}{4 	imes 3 	imes 2 	imes 1} = 210$. For $n(P)$, we need $5$-digit numbers whose product of digits is $20$. The prime factorization of $20$ is $2^2 	imes 5$. The possible sets of $5$ digits are: $1)$ ${5, 4, 1, 1, 1}$: The number of arrangements is $\frac{5!}{3!} = 20$. $2)$ ${5, 2, 2, 1, 1}$: The number of arrangements is $\frac{5!}{2!2!} = 30$. Thus, $n(P) = 20 + 30 = 50$. Therefore, $n(S) + n(P) = 210 + 50 = 260$.

Let $S$ denote the set of $4$-digit numbers $abcd$ such that $a > b > c > d$ and $P$ denote the set of $5$-digit numbers having the product of its digits equal to $20$. Then $n(S) + n(P)$ is equal to:

Similar Questions

Let $x$ and $y$ be two $2$-digit numbers such that $y$ is obtained by reversing the digits of $x$. Suppose they also satisfy $x^2-y^2=m^2$ for some positive integer $m$. The value of $x+y+m$ is

Assuming the balls to be identical except for their color,the number of ways in which one or more balls can be selected from $10$ white,$9$ green,and $7$ black balls is:

The difference between an integer $n$ and its cube $n^3$ is always divisible by which of the following?

The number of ways of giving $20$ distinct oranges to $3$ children such that each child gets at least one orange is $............$.

Consider the following two statements:
$I.$ If $n$ is a composite number,then $n$ divides $(n-1)!$.
$II.$ There are infinitely many natural numbers $n$ such that $n^3+2n^2+n$ divides $n!$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module