Binomial Theorem
Grade None
Question:
<p>If for some <span class="math-tex">\(m, n ;{ }^{6} C_{m}+2\left({ }^{6} C_{m+1}\right)+{ }^{6} C_{m+2} \gt { }^{8} C_{3}\)</span> and <span class="math-tex">\({ }^{n-1} P_{3}:{ }^{n} P_{4}=1: 8\)</span>, then <span class="math-tex">\({ }^{n} P_{m+1}+{ }^{n+1} C_{m}\)</span> is equal to</p>
<p style="display:inline">380</p>
<p style="display:inline">376</p>
<p style="display:inline">384</p>
<p style="display:inline">372</p>
Step-by-Step Solution
Key Concept: Apply Pascal's Rule repeatedly to simplify the combination sum into a single binomial coefficient and use the expansion of permutations to isolate variables m and n.
<p>Given,<br />
<span class="math-tex">\({ }^{6} C_{m}+2\left({ }^{6} C_{m+1}\right)+{ }^{6} C_{m+2} \gt { }^{8} C_{3}\)</span><br />
<span class="math-tex">\(\Rightarrow{ }^{7} C_{m+1}+{ }^{7} C_{m+2} \gt { }^{8} C_{3}\)</span><br />
<span class="math-tex">\(\Rightarrow{ }^{8} C_{m+2} \gt { }^{8} C_{3}\)</span><br />
<span class="math-tex">\(\therefore m=2\)</span><br />
And <span class="math-tex">\({ }^{n-1} P_{3}:{ }^{n} P_{4}=1: 8\)</span><br />
<span class="math-tex">\(\frac{(n-1)(n-2)(n-3)}{n(n-1)(n-2)(n-3)}=\frac{1}{8} \Rightarrow n=8\)</span><br />
<span class="math-tex">\(\therefore{ }^{n} P_{m+1}+{ }^{n+1} C_{m}={ }^{8} P_{3}+{ }^{9} C_{2}=372\)</span>.</p>
Correct Answer: D