The value of 20 -21 -1 -22 -2 - - -60 -40 - is

20!+21!1!+22!2!+.....+60!40! =20!(20!0!× 20!+21!1!· 20!+22!2!× 20!+.....+60!40!× 20!) =20!(^20C_0+^21C_1+^22C_2+....+^60C_40) =20!(^21C_0+^21C_1+^22C_2+....+^60 ...

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Standard XII

Mathematics

Factorization Method Form to Remove Indeterminate Form

The value of ...

Question

The value of $20! + \frac{21!}{1!} + \frac{22!}{2!} + . . . . . + \frac{60!}{40!}$ is

20! \cdot^{61} C_{20}

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21! \cdot^{60} C_{20}

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20! \cdot^{61} C_{21}

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21! \cdot^{60} C_{19}

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Solution

The correct option is C $20! \cdot^{61} C_{21}$
$20! + \frac{21!}{1!} + \frac{22!}{2!} + . . . . . + \frac{60!}{40!}$
$= 20! (\frac{20!}{0! \times 20!} + \frac{21!}{1! \cdot 20!} + \frac{22!}{2! \times 20!} + . . . . . + \frac{60!}{40! \times 20!})$
$= 20! (^{20} C_{0} +^{21} C_{1} +^{22} C_{2} + . . . . +^{60} C_{40})$
$= 20! (^{21} C_{0} +^{21} C_{1} +^{22} C_{2} + . . . . +^{60} C_{40})$

We know that,
$(^{21} C_{0} +^{21} C_{1}) =^{22} C_{1}$
$(^{22} C_{1} +^{22} C_{2}) =^{23} C_{2}$ and so on
.
.
.
so,
$(^{60} C_{39} +^{60} C_{40}) =^{61} C_{40} =^{61} C_{21}$

Hence the value of the sum is,
$20! \cdot^{61} C_{21}$

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Similar questions

Table 1
Marks	Frequency	Cumulative Frequency
0-10	2	2
10-20	8	10
20-30	12	22
30-40	18	40
40-50	28	68
50-60	22	90
60-70 70-80	6 4	96 100

Table 2

Class (marks )	Frequency
11-15	2
16-20	3
21-25	3
26-30	5
31-35	6
36-40	6
41-45	3
46-50	2
Total	30

Table 3

Time (in min)	Frequency
15	7
20	15
25	17
30	3
35	14
40	2

Table 4

31	42	49	25	29
40	36	24	29	30
40	46	38	34	26
24	22	39	40	21

Which table shows the cumulative frequency distribution?

Q. The arithmetic mean for the following distribution is

53

. Find the value of

p

$C l a s s$	$0 - 20$	$20 - 40$	$40 - 60$	$60 - 80$	$80 - 100$
$F r e q u e n c y$	$15$	$18$	$21$	$p$	$17$

Q. Find the missing frequencies if the mean of the given data is

53

Age in years	$0 - 20$	$20 - 40$	$40 - 60$	$60 - 80$	$80 - 100$	Total
Number of People	$15$	$F_{1}$	$21$	$F_{2}$	$17$	$100$

Q. Find the value of 20+21+22+23+ .......+50

Q. Fill the value in blank

22 x 12	10	___
20	200	40
2	20	20

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The value of 20!+21!1!+22!2!+.....+60!40! is

The value of $20! + \frac{21!}{1!} + \frac{22!}{2!} + . . . . . + \frac{60!}{40!}$ is