Permutations & Combinations
Combinations
Grade 11
Question:
<p>A shopkeeper places before you 41 different toys out which 20 toys are to be purchased. Suppose \(m\) = number of ways in which 20 toys can be purchased without any restriction and \(n\) = number of ways in which a particular toy is to be always included in each selection of 20 toys, then \((m - n)\) can be expressed as:</p>
<p>(a) \(\dfrac{2^{10}}{20!}(1 \cdot 3 \cdot 5 \cdots 39)\)</p>
<p>(b) \(\dfrac{2^{20}(1 \cdot 3 \cdot 5 \cdots 19)}{10!}\)</p>
<p>(c) \(\displaystyle\prod_{r=0}^{19}\left(\dfrac{4r+2}{20-r}\right)\)</p>
<p>(d) \(\left(\dfrac{21}{1}\right)\left(\dfrac{22}{2}\right)\left(\dfrac{23}{3}\right)\cdots\left(\dfrac{40}{20}\right)\)</p>
Step-by-Step Solution
Key Concept: The difference (m - n) represents selections where a particular toy is NOT included; this equals C(40, 20) since we're choosing 20 from the remaining 40 toys when one specific toy is excluded.
<p><strong>Step 1:</strong> Calculate m (unrestricted selections)</p><p>m = C(41, 20) = ways to choose 20 toys from 41 different toys</p><p><strong>Step 2:</strong> Calculate n (particular toy always included)</p><p>If one particular toy must be included, we choose the remaining 19 toys from the remaining 40 toys</p><p>n = C(40, 19)</p><p><strong>Step 3:</strong> Find (m - n)</p><p>m - n = C(41, 20) - C(40, 19)</p><p>Using the identity: C(n, r) - C(n-1, r-1) = C(n-1, r)</p><p>m - n = C(40, 20)</p><p><strong>Alternative insight:</strong> (m - n) represents selections where the particular toy is NOT included, which is exactly C(40, 20) (choosing 20 from remaining 40 toys)</p><p>∴ Answer: ACD</p>
Correct Answer: ACD