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Chapter 11: Problem 2

Express each row of Pascal's triangle using combinations. Leave each term in the form \(_{n} C_{r}\). a) \(1 \quad 2 \quad 1\) b) \(1 \quad 4 \quad 6 \quad 4 \quad 1\) c) \(1 \quad 7 \quad 21 \quad 35 \quad 35 \quad 21 \quad 7 \quad 1\)

Short Answer

Expert verified
a) \( _{2} C_{0} \), \( _{2} C_{1} \), \( _{2} C_{2} \) b) \( _{4} C_{0} \), \( _{4} C_{1} \), \( _{4} C_{2} \), \( _{4} C_{3} \), \( _{4} C_{4} \) c) \( _{7} C_{0} \), \( _{7} C_{1} \), \( _{7} C_{2} \), \( _{7} C_{3} \), \( _{7} C_{4} \), \( _{7} C_{5} \), \( _{7} C_{6} \), \( _{7} C_{7} \)

Step by step solution

01

Identify the Row Number

Pascal's triangle starts from row 0 at the top. Identify the row numbers for the given sequences. For sequence (a) it corresponds to the 2nd row, for (b) the 4th row, and for (c) the 7th row.
02

Understand the Symbol

The notation used here \(_{n} C_{r}\) represents a combination which is calculated as \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] where \( n \) is the row number and \( r \) is the position in the row starting from 0.
03

Exercise Part (a)

For \(1 \ 2 \ 1\), which is the 2nd row, express each term in the form \( _{2} C_{r} \): - 1 = \( _{2} C_{0} \) - 2 = \( _{2} C_{1} \) - 1 = \( _{2} C_{2} \)
04

Exercise Part (b)

For \( 1 \ 4 \ 6 \ 4 \ 1 \), which is the 4th row, express each term in the form \( _{4} C_{r} \): - 1 = \( _{4} C_{0} \) - 4 = \( _{4} C_{1} \) - 6 = \( _{4} C_{2} \) - 4 = \( _{4} C_{3} \) - 1 = \( _{4} C_{4} \)
05

Exercise Part (c)

For \( 1 \ 7 \ 21 \ 35 \ 35 \ 21 \ 7 \ 1 \), which is the 7th row, express each term in the form \( _{7} C_{r} \): - 1 = \( _{7} C_{0} \) - 7 = \( _{7} C_{1} \) - 21 = \( _{7} C_{2} \) - 35 = \( _{7} C_{3} \) - 35 = \( _{7} C_{4} \) - 21 = \( _{7} C_{5} \) - 7 = \( _{7} C_{6} \) - 1 = \( _{7} C_{7} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinations
To fully understand Pascal's Triangle, it's crucial to grasp the concept of combinations. A combination is a way to select items from a larger pool, where the order does not matter. It's different from permutations where order is important.
To calculate combinations, we use the notation \(_{n}C_{r}\). This represents the number of ways to choose \(r\) items from \(n\) items without caring about the order.
  • For example, \( _{4}C_{2} \) means choosing 2 items out of 4.
Understanding combinations is essential because it's the foundation of Pascal's Triangle and is widely used in probability and statistics.
Binomial Coefficients
Binomial coefficients are closely related to combinations. In fact, they are represented in the same way using \(_{n}C_{r}\). The binomial coefficient is calculated using the formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
In this formula:
  • \(n!\) (read as \(n\) factorial) is the product of all positive integers up to \(n\).
  • \(r!\) is the factorial of \(r\).
  • \((n-r)!\) is the factorial of \((n-r)\).
These coefficients are used in the expansion of binomial expressions. For example, the coefficients in the expansion of \((a+b)^n\) are given by the binomial theorem, and they correspond to the rows of Pascal's Triangle.
Each element in a row of Pascal's Triangle is a binomial coefficient. This means that the elements of Pascal's Triangle can be expressed as \(_{n}C_{r}\), where \(n\) is the row number, and \(r\) is the position in that row.
Factorial Notation
The factorial notation is integral to understanding combinations and binomial coefficients. The notation \(n!\) (read as \(n\) factorial) is defined as the product of all positive integers from 1 to \(n\).
  • For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
Factorial notation helps simplify the calculation of combinations and binomial coefficients because it succinctly represents the product of a series of descending natural numbers.
When calculating combinations, you use factorials to determine the number of ways to choose items. Factorials also appear in the denominator of the combination formula to eliminate permutations (the different ways to arrange items), thus focusing on unique selections.
In summary, master factorial notation to easily work with combinations and binomial coefficients in Pascal's Triangle and beyond.

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Most popular questions from this chapter

a) Draw a tree diagram that depicts tossing a coin three times. Use H to represent a head and T to represent a tail landing face up. List the arrangements of heads (H) and tails (T) by the branches of your tree diagram. b) Expand \((\mathrm{H}+\mathrm{T})^{3}\) by multiplying the factors. In the first step write the factors in full. For example, the first term will be HHH. You should have eight different terms. Simplify this arrangement of terms by writing HHH as \(\mathrm{H}^{3},\) and so on. Combine like terms. c) What does HHH or \(\mathrm{H}^{3}\) represent in both part a) and part b)? Explain what 3HHT or \(3 \mathrm{H}^{2} \mathrm{T}\) represents in parts a) and b).

How many even numbers of at least four digits can be formed using the digits \(0,1,2,3,\) and 5 without repetitions?

Describe the cases you could use to solve each problem. Do not solve. a) How many 3 -digit even numbers greater than 200 can you make using the digits \(1,2,3,4,\) and \(5 ?\) b) How many four-letter arrangements beginning with either B or E and ending with a vowel can you make using the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{E}, \mathrm{U},\) and \(\mathrm{G} ?\)

A national organization plans to issue its members a 4 -character ID code. The first character can be any letter other than O. The last 3 characters are to be 3 different digits. If the organization has 25300 members, will they be able to assign each member a different ID code? Explain.

Nine students take a walk on four consecutive days. They always walk in rows of three across. Show how to arrange the students so that each student walks only once in a row with any two other students during the four-day time frame. In other words, no three-across triplets are repeated.
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