Match the following-:
Column- I | Column- II |
(i) The sum of the series is | [A] – 10C5 |
(ii) The coefficient of x53 in is | [B] 100C4 |
(iii) (10C0)2 –(10C1)2 +……. …..–(10C9)2 + (10C10)2 equals | [C] 219+20C10 |
(iv) The value of 95C4 + is | [D] – 100C53 |
Text Solution
Verified by Experts(i) [C]; (ii) [D]; (iii) [A]; (iv) [B]
Ans.
(i) [C]
(ii) [D]
(iii) [A]
(iv) [B]
Sol. (i) 
= 20 C 0 + 20 C 1 +……+ 20 C 10
But, 20 C 0 + 20 C 1 +……+ 20 C 20 = 2 20
Also, 20 C 20 = 1 = 20 C 0 , 20 C 19 = 20 C 1 , 20 C 18 = 20 C 2 etc.
∴ given sum = ( 20 C 0 + 20 C 1 +……+ 20 C 20 )
– ( 20 C 11 +…..+ 20 C 20 )
2 20 + 20 C 10 – ( 20 C 10 + 20 C 9 + ……+ 20 C 0 )
∴ 2 ( 20 C 0 + 20 C 1 +…..+ 20 C 10 )
= 2 20 + 20 C 10
(ii) 
(x –3) 100–r 2 r = ((x–3) +2)
100 = (x –1) 100 = (1 –x) 100
= 
∴ Coeff. of x 53 = (–1) 53 100 C 53 = – 100 C 53
(iii) We have
(1+ x) 10 = 10 C 0 + 10 C 1 x + 10 C 2 x 2 +……+ 10 C 10 x 10 ....(1)Also (1–x)
10 = 10 C 0 – 10 C 1 x + 10 C 2 x 2 +…….…..+ 10 C 10 x 10 ....(2)
Multiplying, we get
(1 –x 2 ) 10 = ( 10 C 0 + 10 C 1 x + 10 C 2 x 2 +……….+ 10 C 10 x 10 ) × ( 10 C 0 – 10 C 1 x + 10 C 2 x 2 +……...+ 10 C 10 x 10) Equating the coefficients of x
10 , we get
10 C 5 (–1) 5 = 10 C 0 10 C 10 – 10 C 1 10 C 9 + 10 C 2 10 C 8 +… ….+ 10 C 10 10 C 0
⇒ – 10 C 5 = ( 10 C 0 ) 2 – ( 10 C 1 ) 2 + ( 10 C 2 ) 2 +………+ ( 10 C 10 ) 2
(iv) 95 C 4 + 
= 95 C 4 + 99 C 3 + 98 C 3 + 97 C 3 + 96 C 3 + 95 C 3
= ( 95 C 4 + 95 C 3 ) + 96 C 3 + 97 C 3 + 98 C 3 + 99 C 3
= ( 96 C 4 + 96 C 3 ) + 97 C 3 + 98 C 3 + 99 C 3
= ( 97 C 4 + 97 C 3 ) + 98 C 3 + 99 C 3
= ( 98 C 4 + 98 C 3 ) + 99 C 3
= 99 C 4 + 99 C 3
= 100 C 4
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