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Q5(i):
Evaluate $frac{n!}{(n-r)!}$, when (i) $n = 6, r = 2$
Solution :
Step 1: Identifying the Mathematical Expression
The problem requires the evaluation of the factorial expression $\frac{n!}{(n-r)!}$ given the parameters $n = 6$ and $r = 2$. This expression represents the formula for calculating permutations of $n$ items taken $r$ at a time, denoted as $P(n, r) = \frac{n!}{(n-r)!}$. [By the Fundamental Principle of Counting]
Step 2: Substitution of Given Variables
Substitute $n = 6$ and $r = 2$ into the expression:
$\frac{6!}{(6-2)!}$
Simplifying the denominator: $6 - 2 = 4$.
The expression becomes: $\frac{6!}{4!}$
Step 3: Expansion of Factorials
Recall the definition of a factorial: $n! = n \times (n-1) \times (n-2) \times \dots \times 1$. [By the definition of the factorial function]
Expand $6!$ until reaching $4!$ to allow for cancellation:
$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$
$6! = 6 \times 5 \times 4!$
Step 4: Algebraic Simplification
Substitute the expanded form back into the quotient:
$\frac{6 \times 5 \times 4!}{4!}$
Cancel the common factor $4!$ from both the numerator and the denominator: [By the Property of Multiplicative Identity]
$6 \times 5 = 30$
Visual Representation of Permutation Calculation
Step 5: Tabular Summary of Computation
| Component | Value |
|---|---|
| Numerator ($n!$) | $720$ |
| Denominator ($(n-r)!$) | $24$ |
| Result | $30$ |
Final Solution: 30
More Questions from Class 11 Mathematics Permutations and Combinations EXERCISE 6.2
CBSE Solutions for Class 11 Mathematics Permutations and Combinations
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