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Magazine Chapter 7: Problem 77
The value of \({ }^{50} C_{4}+\sum_{r=1}^{6}{ }^{56-r} C_{3}\) is (a) \({ }^{55} \mathrm{C}_{4}\) (b) \({ }^{55} C_{3}\) (c) \({ }^{56} \mathrm{C}_{3}\) (d) \({ }^{56} C_{4}\)
Short Answer
Expert verified
(d) \(^{56} C_{4}\) is the value of the expression.
Step by step solution
01
Break Down the Expression
We have the expression \(^{50} C_{4} + \sum_{r=1}^{6} {}^{56-r} C_{3}\). The task is to calculate each part of this expression separately to simplify it.
02
Simplify the Summation
The summation expression is \(\sum_{r=1}^{6} {}^{56-r} C_{3}\). It means we need to evaluate \(^{55} C_{3} + {}^{54} C_{3} + {}^{53} C_{3} + {}^{52} C_{3} + {}^{51} C_{3} + {}^{50} C_{3}\).
03
Use the Identity Formula
We can use the identity for binomial coefficients: \(^{n} C_{k} + {}^{n} C_{k+1} = {}^{n+1} C_{k+1}\). Apply this repeatedly to the terms of the summation.
04
Apply the Identity
Starting from \(^{55} C_{3} + {}^{54} C_{3} = {}^{55} C_{4}\), continue combining: \(^{55} C_{4} + {}^{53} C_{3} = {}^{54} C_{4}, {}^{54} C_{4} + {}^{52} C_{3} = {}^{53} C_{4}, {}^{53} C_{4} + {}^{51} C_{3} = {}^{52} C_{4}\), and finally \(^{52} C_{4} + {}^{50} C_{3} = {}^{51} C_{4}\).
05
Final Identity Application
The result from the previous step, \(^{51} C_{4}\), can be combined with \(^{50} C_{4}\) from the original expression using the identity: \(^{51} C_{4} + {}^{50} C_{4} = {}^{51} C_{5}\).
06
Reconstruct the Final Expression
The expression after applying identities ultimately simplifies to \(^{56} C_{4}\). This represents the final solution for the given expression.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Combinatorics
Combinatorics is a branch of mathematics dealing with combinations, permutations, and counting. It helps us determine how many ways we can select items from a group. In our problem, we are working with binomial coefficients, which are part of combinatorial mathematics. These coefficients, represented as \(^{n} C_{k}\), tell us how many ways we can choose \(k\) elements from a set of \(n\) elements without considering the order.
- The notation \(^{n} C_{k}\) is often read as 'n choose k.'
- Binomial coefficients are used in the expansion of binomial expressions, like \((a + b)^n\).
- Combinatorics enables us to break down complex counting problems into manageable parts.
Summation
Summation is the process of adding a sequence of numbers; in our case, it is used to calculate binomial coefficients over a range. The summation symbol \(\sum\) indicates that we add several terms together. In the problem, we address \[\sum_{r=1}^{6} {}^{56-r} C_{3}\].
- The limits of the summation indicate that \(r\) will start at 1 and increase to 6.
- Each term takes the form \(^{56-r} C_{3}\), showing the combination of 3 items from a constantly decreasing number of items starting from 55.
- This approach simplifies evaluating a sequence of similar problems all at once.
Identity Formula
An identity formula is a mathematical expression that equates two expressions for every possible value of their variables. Here, we use an important identity in binomial coefficients: \[^{n} C_{k} + {}^{n} C_{k+1} = {}^{n+1} C_{k+1}\].
- This identity is instrumental in simplifying the sum of binomial coefficients.
- As seen in the solution, it allows us to combine terms progressively, reducing complexity.
- The pattern becomes clear upon repeated application, collapsing multiple terms into one neat expression.
Simplification in Mathematics
Simplification involves reducing a complex expression to a simpler form while maintaining its value and meaning. It is a fundamental skill in solving mathematical problems effectively. The given exercise exemplifies simplification through the combination of binomial coefficients and identities.
- The steps involve breaking down the original expression and applying identities to combine terms.
- Simplification helps in recognizing more straightforward solutions within seemingly complex problems.
- It often involves recognizing patterns and relationships among the terms.
