If 1P1-2-2P2-3-3P3-15-15P15-qPr-s- 0- s- 1- then q-sCr-s is equal to

∑ _r=1^15r·^rP_r=∑ _r=1^15r· r!=∑ _r=1^15(r+1 1)·r! =∑ _r=1^15(r+1)r! r!=∑ _r=1^15(r+1)! r! T_1=2! 1! T_2=3! 2! T_3=4! 3! ⋮ nbsp; T_15=16! 15! Sum =16! 1= ^1 ...

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Byju's Answer

Standard XII

Mathematics

Sum of Coefficients of All Terms

If 1P1+2·2P2+...

Question

If $^{1} P_{1} + 2 \cdot^{2} P_{2} + 3 \cdot^{3} P_{3} + \dots + 15 \cdot^{15} P_{15} =^{q} P_{r} - s, 0 \leq s \leq 1,$ then $^{q + s} C_{r - s}$ is equal to

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Solution

$15 \sum r = 1 r \cdot^{r} P_{r} = 15 \sum r = 1 r \cdot r! = 15 \sum r = 1 (r + 1 - 1) \cdot r!$
$= 15 \sum r = 1 (r + 1) r! - r! = 15 \sum r = 1 (r + 1)! - r!$

$T_{1} = 2! - 1!$
$T_{2} = 3! - 2!$
$T_{3} = 4! - 3!$
$⋮$
$T_{15} = 16! - 15!$
Sum $= 16! - 1 =^{16} p_{16} - 1$
$q = 16, r = 16, s = 1$
$∴^{q + s} C_{r - s} =^{17} C_{15}$
$= 136$

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If 1P1+2⋅2P2+3⋅3P3+⋯+15⋅15P15=qPr−s, 0≤s≤1, then q+sCr−s is equal to

15∑r=1r⋅rPr=15∑r=1r⋅r!=15∑r=1(r+1−1)⋅r! =15∑r=1(r+1)r!−r!=15∑r=1(r+1)!−r! T1=2!−1! T2=3!−2! T3=4!−3! ⋮ T15=16!−15! Sum =16!−1= 16p16−1 q=16, r=16, s=1 ∴q+sCr−s= 17C15 =136

If $^{1} P_{1} + 2 \cdot^{2} P_{2} + 3 \cdot^{3} P_{3} + \dots + 15 \cdot^{15} P_{15} =^{q} P_{r} - s, 0 \leq s \leq 1,$ then $^{q + s} C_{r - s}$ is equal to