Class 11: Combinations – Very Short Answer Questions

$\displaystyle \textbf{Question 1. }\text{Write }\sum_{r=0}^{m}{}^{n+r}C_r\text{ in the simplified form.}$
$\displaystyle \text{Answer:}$
$\displaystyle \sum_{r=0}^{m}{}^{n+r}C_r={}^{n}C_0+{}^{n+1}C_1+{}^{n+2}C_2+\cdots+{}^{n+m}C_m$
$\displaystyle \text{Using the identity }{}^{p}C_q+{}^{p}C_{q+1}={}^{p+1}C_{q+1},$
$\displaystyle \sum_{r=0}^{m}{}^{n+r}C_r={}^{n+m+1}C_m$
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$\displaystyle \textbf{Question 2. }\text{If }{}^{35}C_{n+7}={} ^{35}C_{4n-2},\text{ then write the values of }n.$
$\displaystyle \text{Answer:}$
$\displaystyle {}^{35}C_{n+7}={} ^{35}C_{4n-2}$
$\displaystyle \therefore n+7=4n-2\text{ or }n+7+4n-2=35$
$\displaystyle 3n=9\text{ or }5n+5=35$
$\displaystyle n=3\text{ or }n=6$
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$\displaystyle \textbf{Question 3. }\text{Write the number of diagonals of an }n\text{-sided polygon.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Number of diagonals}={}^nC_2-n$
$\displaystyle =\frac{n(n-1)}{2}-n$
$\displaystyle =\frac{n(n-3)}{2}$
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$\displaystyle \textbf{Question 4. }\text{Write the expression }{}^nC_{r+1}+{}^nC_{r-1}+2\times{}^nC_r\text{ in the simplest form.}$
$\displaystyle \text{Answer:}$
$\displaystyle {}^nC_{r+1}+{}^nC_r+{}^nC_r+{}^nC_{r-1}$
$\displaystyle ={}^{n+1}C_{r+1}+{}^{n+1}C_r$
$\displaystyle ={}^{n+2}C_{r+1}$
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$\displaystyle \textbf{Question 5. }\text{Write the value of }\sum_{r=1}^{6}{}^{56-r}C_3+{}^{50}C_4.$
$\displaystyle \text{Answer:}$
$\displaystyle \sum_{r=1}^{6}{}^{56-r}C_3+{}^{50}C_4$
$\displaystyle ={}^{55}C_3+{}^{54}C_3+\cdots+{}^{50}C_3+{}^{50}C_4$
$\displaystyle =\left({}^{56}C_4-{}^{50}C_4\right)+{}^{50}C_4$
$\displaystyle ={}^{56}C_4$
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$\displaystyle \textbf{Question 6. }\text{There are }3\text{ letters and }3\text{ directed envelopes. Write the number of}$
$\displaystyle \text{ways in which no letter is put in the correct envelope.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Required number is the number of derangements of }3\text{ objects.}$
$\displaystyle !3=3!\left(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}\right)$
$\displaystyle =6\left(1-1+\frac{1}{2}-\frac{1}{6}\right)=2$
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$\displaystyle \textbf{Question 7. }\text{Write the maximum number of points of intersection of }8 \\ \text{ straight lines in a plane.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Maximum number of points of intersection}={}^8C_2$
$\displaystyle =\frac{8\times7}{2}=28$
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$\displaystyle \textbf{Question 8. }\text{Write the number of parallelograms that can be formed from a set of}$
$\displaystyle \text{four parallel lines intersecting another set of three parallel lines.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Choose }2\text{ lines from }4\text{ parallel lines and }2\text{ lines from }3\text{ parallel lines.}$
$\displaystyle \text{Number of parallelograms}={}^4C_2\times{}^3C_2$
$\displaystyle =6\times3=18$
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$\displaystyle \textbf{Question 9. }\text{Write the number of ways in which }5\text{ red and }4\text{ white balls can be}$
$\displaystyle \text{drawn from a bag containing }10\text{ red and }8\text{ white balls.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Number of ways}={} ^{10}C_5\times{}^8C_4$
$\displaystyle =252\times70=17640$
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$\displaystyle \textbf{Question 10. }\text{Write the number of ways in which }12\text{ boys may be divided into} \\ \text{three groups of }4\text{ boys each.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Required number of ways}=\frac{12!}{4!\,4!\,4!\,3!}$
$\displaystyle =5775$
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$\displaystyle \textbf{Question 11. }\text{Write the total number of words formed by }2\text{ vowels and }3$
$\displaystyle \text{ consonants taken from }4\text{ vowels and }5\text{ consonants.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Choose }2\text{ vowels from }4\text{ and }3\text{ consonants from }5.$
$\displaystyle \text{Number of selections}={}^4C_2\times{}^5C_3$
$\displaystyle \text{Each selection has }5\text{ letters, which can be arranged in }5!\text{ ways.}$
$\displaystyle \therefore \text{Total number of words}={}^4C_2\times{}^5C_3\times5!$
$\displaystyle =6\times10\times120=7200$
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