If the number of 5-element subsets of the set | vedclass

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(C) The total number of $5$-element subsets of a set with $20$ elements is given by $^{20}C_5$.The number of $5$-element subsets containing a specific

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(C) The total number of $5$-element subsets of a set with $20$ elements is given by $^{20}C_5$.The number of $5$-element subsets containing a specific element $a_4$ is equivalent to choosing $4$ more elements from the remaining $19$ elements,which is $^{19}C_4$.According to the problem,$^{20}C_5 = k 	imes ^{19}C_4$.Using the formula $^{n}C_r = \frac{n}{r} 	imes ^{n-1}C_{r-1}$,we have:$^{20}C_5 = \frac{20}{5} 	imes ^{19}C_4 = 4 	imes ^{19}C_4$.Comparing this with $k 	imes ^{19}C_4$,we get $k = 4$.

If the number of $5$-element subsets of the set $A = \{a_1, a_2, \dots, a_{20}\}$ containing $20$ distinct elements is $k$ times the number of $5$-element subsets containing the element $a_4$,then what is the value of $k$?

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$^nC_r + ^{n-1}C_r + ...... + ^rC_r =$

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