Let m, n be two distinct integers chosen rand | vedclass

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

Question

(A) The total number of ways to choose two distinct integers $m$ and $n$ from the set $\{0, 1, 2, \ldots, 99\}$ is given by $\binom{100}{2} = \frac{10

Vedclass · Accepted Answer

$(0, 0.25]$

Vedclass · Answer

(A) The total number of ways to choose two distinct integers $m$ and $n$ from the set $\{0, 1, 2, \ldots, 99\}$ is given by $\binom{100}{2} = \frac{100 	imes 99}{2} = 4950$.We want $4^m + 4^n + 3 \equiv 0 \pmod{5}$.Since $4 \equiv -1 \pmod{5}$,we have $4^m + 4^n + 3 \equiv (-1)^m + (-1)^n + 3 \pmod{5}$.For this expression to be divisible by $5$,we need $(-1)^m + (-1)^n + 3 \equiv 0 \pmod{5}$,which implies $(-1)^m + (-1)^n \equiv -3 \equiv 2 \pmod{5}$.Since $(-1)^m$ and $(-1)^n$ can only be $1$ or $-1$,the only way for their sum to be $2$ is if $(-1)^m = 1$ and $(-1)^n = 1$.This means both $m$ and $n$ must be even integers.In the set $\{0, 1, 2, \ldots, 99\}$,there are $50$ even integers $(	ext{i.e.}, 0, 2, 4, \ldots, 98)$.The number of ways to choose two distinct even integers is $\binom{50}{2} = \frac{50 	imes 49}{2} = 1225$.The probability is $P = \frac{1225}{4950} = \frac{49}{198} \approx 0.24747$.Since $0.24747 \in (0, 0.25]$,the correct interval is $(0, 0.25]$.

Let $m, n$ be two distinct integers chosen randomly from the set $\{0, 1, 2, \ldots, 99\}$. Then,the probability that $4^m + 4^n + 3$ is divisible by $5$ lies in the interval

Similar Questions

If the probability that a randomly chosen $6$-digit number formed by using digits $1$ and $8$ only is a multiple of $21$ is $p$,then $96p$ is equal to

Seven chits are numbered $1$ to $7$. Three are drawn one by one with replacement. The probability that the least number on any selected chit is $5$ is:

Find the probability that when a hand of $7$ cards is drawn from a well-shuffled deck of $52$ cards,it contains all $4$ Kings.

If two sections of strengths $30$ and $45$ are formed from $75$ students who are admitted in a school,then the probability that two particular students are always together in the same section is

There are $8$ boys and $7$ girls in a classroom. If the names of all those children are written on paper slips and $3$ slips are drawn at random from them,then the probability of getting the names of one boy and two girls or one girl and two boys is

Vedclass Test Series

Exam Paper Generator

Online Exam Module