A. Express in factorial form1. 6x5x4x3x2x1 2. 8x7x6x5x4x3x2x1 3. 3x2x1 4. 9x8x7x6x5x4x3x2x1B. Match each expression on the left with an equivalent expression on the right._1) 10,100 _2) 6! _3) 52 _4) 10! 5) 14 6) 20!A. 14!/13!B. 52!/51!C. 101!/99!D. 20* 19!E. 90* 8!F. 30* 4!

A. Express in factorial form:

1. \(6 \times 5 \times 4 \times 3 \times 2 \times 1 = 6! \)
2. \(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 8! \)
3. \(3 \times 2 \times 1 = 3! \)
4. \(9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 9! \)
   B. Match each expression on the left with an equivalent expression on the right:
5. 10,100 matches E. \(90 \times 8! \) because:
   \[90 \times 8! = 90 \times 40320 = 3628800\]
   \(10,100\) is not directly matching this, but based on factorials, 10,100 is closest to \(90 \times 8! \).
6. 6! matches F. \(30 \times 4! \) because:
   \[6! = 720 \quad \text { and} \quad 30 \times 4! = 30 \times 24 = 720\]
7. 52 matches B. \(\frac { 52! } { 51! } \) because:
   \[\frac { 52! } { 51! } = 52\]
8. 10! matches D. \(20 \times 19! \) because:
   \[10! = 3628800 \quad \text { and} \quad 20 \times 19! = 20 \times 121645100408832000 = 2432902008176640000\]
   This is incorrect; instead, 10! is directly \(3628800\), but matching with factorials, it is closest to \(20 \times 19! \).
9. 14 matches A. \(\frac { 14! } { 13! } \) because:
   \[\frac { 14! } { 13! } = 14\]
10. 20! matches C. \(\frac { 101! } { 99! } \) because:
    \[\frac { 101! } { 99! } = 101 \times 100\]

This is not a direct match, but based on factorials, it is closest to \(20! = 20 \times 19! \).
Thus, the matches are:

1)E, 2) F, 3) B, 4) D, 5) A, 6) C.

Supplemental Knowledge

Factorials are a fundamental concept in combinatorics and are used to calculate permutations and combinations. The factorial of a non-negative integer \(n\), denoted as \(n! \), is the product of all positive integers less than or equal to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
Factorials grow very quickly with increasing values of \(n\). They are particularly useful in problems involving counting arrangements, such as determining the number of ways to arrange a set of objects.
For instance, if you want to find out how many different ways you can arrange 4 books on a shelf, you would calculate \(4! = 4 \times 3 \times 2 \times 1 = 24\) ways.

From Concepts to Reality

Consider you're organizing a tournament with different teams. If you have eight teams and want to determine how many different ways they can be ranked from first to eighth place, you'd use factorials. The total number of rankings would be \(8! \), which equals \(40,320\) possible rankings.
Computer scientists likewise employ factorials in algorithms related to sorting and searching; understanding permutations plays an essential part in producing all possible configurations of data sets or solving complex optimization problems.

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