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How many changes can be rung with a peal of 5 bells.

Answer Verified Verified
Hint: Start from 1 bell and take the calculation up to 5 bells, first find the change of variation of 1 bell, find the change of variation of 2 bells, find the change of variation of 3 bells and so on, end this process when someone rang the bell 5 times in the same pattern.

Complete step by step answer:
We have given the $5$ bells about we have to find out the changes which can be rung.
When a bell rung only 1 time, then there will be only one change of variation, which can be written as,
$1! = 1$
When a bell rung only 2 times, then there will be only two changes of variation, which can be written as,
$
  2! = 2 \times 1 \\
   = 2 \\
$
When a bell rung only 3 times, then there will be only six changes of variation, which can be written as,
$
  3! = 3 \times 2 \times 1 \\
   = 6 \\
$
When a bell rung only 4 times, then there will be only twenty-four changes of variation, which can be written as,
$
  4! = 4 \times 3 \times 2 \times 1 \\
   = 24 \\
$
When a bell rung only 5 times, then there will be only 120 changes of variation, which can be written as,
$
  5! = 5 \times 4 \times 3 \times 2 \times 1 \\
   = 120 \\
$

$\therefore$ When a bell rang 5 times, 120 changes of variations were formed.

Note:
Use the concept of periodical repetition and the concept of factorial notation to simplify the calculations. Write the first and second numbers in factorial form instead of simple numbers, otherwise anyone may be confused if factorial notation is not there in the first and second numbers.
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