\(\displaystyle\sum_{r=1}^{m}\)n+rCr Write ∑n+rCr r∈(0,m) in the

1.	\(\displaystyle\sum_{r=1}^{m}\)n+rCr Write ∑n+rCr r∈(0,m) in the simplified form.
Answer» Given, ∑^n+rC_r r∈(0,m) We know : ⁿC_r + ⁿC_r-1 =ⁿ⁺¹C_{r ....}(1) \(\sum_{r=0}^{m} \) ^n+rC_r= ⁿC₀ + ⁿ⁺¹C₁ + ⁿ⁺²C₂ + ⁿ⁺³C₃ + . . . . . . + ^n+mC_n+1 \(\sum_{r=0}^{m} \)^n+rC_r = ⁿ⁺¹C₀ + ⁿ⁺¹C₁ + ⁿ⁺²C₂ + ⁿ⁺³C₃ + . . . . . . +^n+mC_n+1 ⇒ (ⁿC₀ = ⁿ⁺¹C₀) Using equation (1), \(\sum_{r=0}^{m} \)^n+rC_r=ⁿ⁺²C₁ + ⁿ⁺²C₂ + ⁿ⁺³C₃ + . . . . . . + ^n+mC_n+1 \(\sum_{r=0}^{m} \)^n+rC_r= ⁿ⁺³C₂ + ⁿ⁺³C₃ + . . . . . . + ^n+mC_n+1 Proceeding in the same way : \(\sum_{r=0}^{m} \)^n+rC_r= ^n+mC_m-1 + ^n+mC_m = ^n+m+1C_n+1 \(\sum_{r=0}^{m} \)^n+rC_r= ^n+m+1C_n+1

\(\displaystyle\sum_{r=1}^{m}\)n+rCr Write ∑n+rCr r∈(0,m) in the simplified form.

Answer»

Given,

∑^n+rC_r r∈(0,m)

We know :

ⁿC_r + ⁿC_r-1 =ⁿ⁺¹C_{r ....}(1)

\(\sum_{r=0}^{m} \) ^n+rC_r= ⁿC₀ + ⁿ⁺¹C₁ + ⁿ⁺²C₂ + ⁿ⁺³C₃ + . . . . . . + ^n+mC_n+1

\(\sum_{r=0}^{m} \)^n+rC_r = ⁿ⁺¹C₀ + ⁿ⁺¹C₁ + ⁿ⁺²C₂ + ⁿ⁺³C₃ + . . . . . . +^n+mC_n+1

⇒ (ⁿC₀ = ⁿ⁺¹C₀)

Using equation (1),

\(\sum_{r=0}^{m} \)^n+rC_r=ⁿ⁺²C₁ + ⁿ⁺²C₂ + ⁿ⁺³C₃ + . . . . . . + ^n+mC_n+1

\(\sum_{r=0}^{m} \)^n+rC_r= ⁿ⁺³C₂ + ⁿ⁺³C₃ + . . . . . . + ^n+mC_n+1

Proceeding in the same way :

\(\sum_{r=0}^{m} \)^n+rC_r= ^n+mC_m-1 + ^n+mC_m = ^n+m+1C_n+1

\(\sum_{r=0}^{m} \)^n+rC_r= ^n+m+1C_n+1