The sum -sum i -0- m pmatrix 1011ilend pmatrix pmatrix 20 -m - iend pmatrix - where p - q -0- if -p - q - is maximum when m isA- 15B- 5C- 10D- 20

The correct option is A 15For m = 5, ∑_r=0^5[ 10; i ][ 20; 5 i ] =[ 10; 0 ][ 20; 5 ] + [ 10; 1 ][ 20; 4 ] + …… + [ 10; 5 ][ 20; 0 ] for m = 10, ∑_i=0^5 ...

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Byju's Answer

Standard XII

Mathematics

Addition and Subtraction of a Matrix

The sum 'sum ...

Question

The sum $\sum_{i = 0}^{m} (\begin{matrix} 10 i \end{matrix}) (\begin{matrix} 20 m - i \end{matrix})$ , $(w h e r e (\frac{p}{q} = 0 i f p < q))$ , is maximum when m is

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Solution

The correct option is B 15
For $m = 5, \sum_{r = 0}^{5} (\begin{matrix} 10 i \end{matrix}) (\begin{matrix} 20 5 - i \end{matrix})$
$= (\begin{matrix} 100 \end{matrix}) (\begin{matrix} 205 \end{matrix}) + (\begin{matrix} 101 \end{matrix}) (\begin{matrix} 204 \end{matrix}) + \dots \dots + (\begin{matrix} 105 \end{matrix}) (\begin{matrix} 200 \end{matrix})$
for $m = 10, \sum_{i = 0}^{5} (\begin{matrix} 10 i \end{matrix}) (\begin{matrix} 20 10 - i \end{matrix})$
$= (\begin{matrix} 100 \end{matrix}) (\begin{matrix} 2010 \end{matrix}) + (\begin{matrix} 101 \end{matrix}) (\begin{matrix} 209 \end{matrix}) + (\begin{matrix} 102 \end{matrix}) (\begin{matrix} 208 \end{matrix}) + \dots \dots + (\begin{matrix} 1010 \end{matrix}) (\begin{matrix} 200 \end{matrix})$
for $m = 15, \sum_{i = 0}^{15} (\begin{matrix} 10 i \end{matrix}) (\begin{matrix} 20 15 - i \end{matrix})$
$= (\begin{matrix} 100 \end{matrix}) (\begin{matrix} 2015 \end{matrix}) + (\begin{matrix} 101 \end{matrix}) (\begin{matrix} 2014 \end{matrix}) + (\begin{matrix} 102 \end{matrix}) (\begin{matrix} 2013 \end{matrix}) + \dots \dots + (\begin{matrix} 1010 \end{matrix}) (\begin{matrix} 205 \end{matrix})$
and for m = 20, $\sum_{i = 0}^{20} (\begin{matrix} 10 i \end{matrix}) (\begin{matrix} 20 20 - i \end{matrix})$
$= (\begin{matrix} 100 \end{matrix}) (\begin{matrix} 2020 \end{matrix}) + (\begin{matrix} 101 \end{matrix}) (\begin{matrix} 2019 \end{matrix}) + \dots \dots + (\begin{matrix} 1010 \end{matrix}) (\begin{matrix} 2010 \end{matrix})$
Clearly, the sum is maximum for m = 15
Note that $^{10} C_{r}$ is maximum for r = 5 and $^{20} C_{r}$ is maximum for r = 10. Note that the single term $^{10} C_{5} \times^{20} C_{10}$ (in case m = 15) is greater than the sum
$^{10} C_{0}^{20} C_{10} +^{10} C_{1}^{20} C_{9} +^{10} C_{2}^{20} C_{8} + \dots \dots$
$^{10} C_{8}^{20} C_{2} +^{10} C_{9}^{20} C_{1} +^{10} C_{10}^{20} C_{0}$ (in case m = 10).
Also the sum in case m = 10 is same as that in case m = 20

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Similar questions

Q. The sum

\sum_{i = 0}^{m} (_{i}^{10}) (_{m - i}^{20})

, (where

(_{q}^{p})

= 0 if

p < q)

is maximum when

m

Q. Assertion :The sum of

m \sum i = 0 (\begin{matrix} 10 i \end{matrix}) (\begin{matrix} 20 m - i \end{matrix})

will be maximum if

m = 15

Reason: The expression

^{m} C_{r}

attain maximum value if

r = \frac{m}{2} .

Q. For nonnegative integers

s

and

r,

let

(\begin{matrix} s r \end{matrix}) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} \frac{s!}{r! (s - r)!} & if r \leq s, 0 & if r > s . \end{matrix}

For positive integers

m

and

n,

let

g (m, n) = m + n \sum p = 0 \frac{f (m, n, p)}{(\begin{matrix} n + p p \end{matrix})}

where for any nonnegative integer

p,

f (m, n, p) = p \sum i = 0 (\begin{matrix} m i \end{matrix}) (\begin{matrix} n + i p \end{matrix}) (\begin{matrix} p + n p - i \end{matrix}) .

Then which of the following statements is/are TRUE?

Q. Let

m, n \in N

and

gcd (2, n) = 1.

30 (\begin{matrix} 300 \end{matrix}) + 29 (\begin{matrix} 301 \end{matrix}) + \dots + 2 (\begin{matrix} 3028 \end{matrix}) + 1 (\begin{matrix} 3029 \end{matrix}) = n \cdot 2^{m},

then

n + m

is equal to

(Here (\begin{matrix} n k \end{matrix}) =^{n} C_{k})

Q. If

(\begin{matrix} p q \end{matrix})

=0 for

p < q

q ϵ W

then

\infty \sum r = 0

(\begin{matrix} n 2 r \end{matrix})

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The sum ∑mi=0(10i)(20m−i), (where (pq=0 if p<q)), is maximum when m is

The sum $\sum_{i = 0}^{m} (\begin{matrix} 10 i \end{matrix}) (\begin{matrix} 20 m - i \end{matrix})$ , $(w h e r e (\frac{p}{q} = 0 i f p < q))$ , is maximum when m is