Question

The value of the expression ^k–1C_k–1 +^kC_k–1 + . . . + ^{n + k – 2}C_{k – 1} is

• A. ^{n + k – 1}C_{k + 1}
• B. ^{n + k – 1}C_{k – 1}
• C. ^{n+ k}C_k
• D. None of these

Answer 1

Answer: (B)

^{n + k – 1}C_{k – 1}

Answer 2

We have ^k–1C_k–1 + ^kC_k-1 + . . . + ^n+k–2C_k–1

= ^kC_k + ^kC_k–1 + ^k+1C_k–1 . . . + ^n+k–2C_k–1 [since ^kC_k = 1

= ^{k 1}C_k–1]

= ^k+1C_k + ^k+1C_k-1 +. . . . + ^n+k–2C_K–1 [ since ^kC_k + ^kC_k-1

= ^k+1C_k]

= ^k+2C_k + . . . . + ^n+k–2C_k–1 = . . . = ^n+k–2C_k + ^n+k–2C_k–1 =^n+k–1C_k—1