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
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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.
- 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.