If
\((^{40}C_0) + (^{41}C_1) + (^{42}C_2) + ...... + (^{60}C_{20}) \frac{m}{n} ^{60}C_{20}\)
m and n are coprime, then m + n is equal to _____.
If
\((^{40}C_0) + (^{41}C_1) + (^{42}C_2) + ...... + (^{60}C_{20}) \frac{m}{n} ^{60}C_{20}\)
m and n are coprime, then m + n is equal to _____.
Correct Answer: 102
Solution and Explanation
The correct answer is 102
\(^{40}C_0 + ^{41}C_1 + ^{42}C_2 + ...... + ^{60}C_{20}\)
\(= ^{40}C_{40} + ^{41}C_{40} + ^{42}C_{40} + ...... + ^{60}C_{40 }\)
\(= ^{61}C_{41}\)
\(= \frac{61}{41} . ^{60}C_{41} \)
∴ m = 61 , n = 41
Therefore , m + n = 102
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main permutations and combinations Questions
- The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is______
- Five people are distributed in four identical rooms. A room can also contain zero people. Find the number of ways to distribute them.
- The number of ways to distribute the 21 identical apples to three children’s so that each child gets at least 2 apples.
- Let \( \alpha = \frac{(4!)!}{(4!)^{3!}} \) and \( \beta = \frac{(5!)!}{(5!)^{4!}} \). Then:
- Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
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.
- Combination refers to the combination of about n things taken k at a time without any repetition.
- The combination is used for a group of data where the order of data does not matter.