Permutations with Identical Items
The problem asks for the number of distinct ways to arrange a set of objects where some objects are identical. Specifically, we have 6 identical red balls and 5 identical blue balls.
Identify Total Items and Groups
- Total number of balls = 6 (red) + 5 (blue) = 11 balls.
- Number of identical red balls ($n_1$) = 6.
- Number of identical blue balls ($n_2$) = 5.
Formula for Distinct Arrangements
The number of distinct permutations of $n$ objects where there are $n_1$ identical objects of type 1, $n_2$ identical objects of type 2, ..., $n_k$ identical objects of type k is given by the formula:
$ \frac{n!}{n_1! n_2! \cdots n_k!} $In this case, $n = 11$, $n_1 = 6$, and $n_2 = 5$.
Calculate Distinct Ways
Substitute the values into the formula:
$ \frac{11!}{6! 5!} $Expand the factorial terms:
$ \frac{11 \times 10 \times 9 \times 8 \times 7 \times 6!}{6! \times (5 \times 4 \times 3 \times 2 \times 1)} $Cancel out $6!$:
$ \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} $Simplify the expression:
$ = \frac{11 \times 10 \times 9 \times 8 \times 7}{120} $Perform cancellations:
$ = 11 \times \frac{10}{5 \times 2} \times \frac{9}{3} \times \frac{8}{4} \times 7 $ $ = 11 \times 1 \times 3 \times 2 \times 7 $ $ = 11 \times 42 $ $ = 462 $Result
There are 462 distinct ways to arrange the 6 red balls and 5 blue balls in a row.
Important Questions from Permutation and Combination (Notes)
- In how many ways can 10 men be divided into two groups of 4 men and 6 men?
- Out of 5 consonants and 4 vowels, how many words of 3 consonants and 3 vowels can be made?
- How many 5-digit numbers can be formed from the digits 0, 2, 3, 4, 6, 7 and 9, using each at most once, which are divisible by 5?
- In how many distinguishable ways can the letters of the word CHANCE be arranged?
- From a group of 40 players, a cricket team of 11 players is chosen. Then, one of the eleven is chosen as the captain of the team. The total number of ways this can be done is
[$\binom{m}{n}$ below means the number of ways $n$ objects can be chosen from $m$ objects]
