The value of the expression \(^{47}C_4+ \displaystyle \sum^5_{j = 1} , ^{52-j}C_3\) is
- $^{47}C_5$
- $^{52}C_5$
- $^{52}C_4$
- None of these
The Correct Option is C
Solution and Explanation
Given expression;
\(^{47}C_4+ \displaystyle \sum^5_{j = 1} , ^{52-j}C_3\)
\(= ^{47}C_4+ ^{51}C_3 + ^{50}C_3 + ^{49}C_3+ ^{48}C_3 + ^{47}C_3\)
\(= ( ^{47}C_4+ ^{47}C_3 )+ ^{48}C_3 + ^{49}C_3+ ^{50}C_3 + ^{51}C_3\)
\(20mm [using ^nC_r + ^nC_{r-1} = ^{n+1}C_r]\)
\(= ( ^{48}C_4+ ^{48}C_3 )+ ^{49}C_3 + ^{50}C_3 + ^{51}C_3\)
\(= ( ^{49}C_4+ ^{49}C_3 )+ ^{50}C_3 + ^{51}C_3\)
\(= ( ^{50}C_4+ ^{50}C_3 )+ ^{51}C_3\)
\(= ^{51}C_4+ ^{51}C_3 = ^{52}C_4\)
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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.