The value of r0- r- 30 is 20 C r 10 C 0 - 20 C r-1 10 C 1 - 20 C 0 10 C ris least- is

The correct option is A 0 From the paragraph the sum of the above equation will be #160; ^30C_r .Now it is least when ^30C_r=1= ^30C_0 Hence # ...

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Binomial Coefficients

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Question

The value of $r (0 \leq r \leq 30)$ is
${^{20} C}_{r} {^{10} C}_{0} + {^{20} C}_{r - 1} {^{10} C}_{1} + . . . . + {^{20} C}_{0} {^{10} C}_{r}$
is least, is

0

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r

S = {^{20} C}_{r} {^{10} C}_{0} + {^{20} C}_{r - 1} {^{10} C}_{1} + . . . . + {^{20} C}_{0} {^{10} C}_{r}

r

S =^{20} C_{r}^{10} C_{0} +^{20} C_{r - 1}^{10} C_{1} . . . . . +^{20} C_{0}^{10} C_{r}

c_{0}, c_{1}, c_{2}, . . . . . . . c_{n}

{(1 + x)}^{n}

c_{0} c_{r} + c_{1} c_{r + 1} + c_{2} c_{r + 2} + . . . . + c_{n - r} c_{n} = \frac{| 2 n - - -}{| n - r - ---- - | n + r - ---- -}

^{40} C_{r} .^{60} C_{0} +^{40} C_{r - 1} .^{60} C_{1} + . . .

r = 50

^{2 n} C_{r}

r = n

r = 0, 1, 2, . . . ., n

C_{0} \cdot C_{r} + C_{1} \cdot C_{r + 1} + C_{2} \cdot C_{r + 2} + . . . . + C_{n - r} \cdot C_{n} =^{2 n} C_{(n + r)}

C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + . . . . . . + C_{n}^{2} =^{2 n} C_{n}

C_{0} \cdot C_{1} + C_{1} \cdot C_{2} + C_{2} \cdot C_{3} + . . . . . + C_{n - 1} \cdot C_{n} =^{2 n} C_{n + 1}

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