MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Combination

Write the val...

Question

Write the value of $\sum_{r = 1}^{6}^{56 - r} C_{3} +^{50} C_{4} .$

Open in App

Solution

$\sum_{r = 1}^{6}^{56 - r} C_{3} +^{50} C_{4} .$

$=^{56 - 1} C_{3} +^{56 - 2} C_{3} +^{56 - 3} C_{3} +^{56 - 4} C_{4} +^{56 - 5} C_{5} +^{56 - 6} C_{6} +^{50} C_{4} =^{55} C_{3} +^{54} C_{3} +^{53} C_{3} +^{52} C_{3} +^{51} C_{3} +^{50} C_{3} +^{50} C_{4}$

$=^{56 - 1} C_{3} +^{56 - 2} C_{3} +^{56 - 3} C_{3} +^{56 - 4} C_{4} +^{56 - 5} C_{5} +^{56 - 6} C_{6} +^{50} C_{4} =^{55} C_{3} +^{54} C_{3} +^{53} C_{3} +^{52} C_{3} +^{51} C_{3} +^{50} C_{3} +^{50} C_{4} =^{55} C_{3} +^{54} C_{3} +^{53} C_{3} +^{52} C_{3} + (^{50} C_{3} +^{51} C_{4})$
$=^{55} C_{3} +^{54} C_{3} +^{53} C_{3} +^{52} C_{3} +^{51} C_{3} + (^{51} C_{3} +^{51} C_{4})$ ${U s i n g^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}}$
$=^{55} C_{3} + =^{54} C_{3} + (^{55} C_{3} +^{54} C_{4}) =^{55} C_{3} + (^{54} C_{3} +^{55} C_{4}) = (^{55} C_{3} +^{55} C_{4}) =^{56} C_{4}$
So,
$\sum_{r = 1}^{6}^{56 - r} C_{3} +^{50} C_{4} =^{56} C_{4}$

Suggest Corrections

thumbs-up

1

Similar questions

Q. The value of

^{50} C_{4} + \sum_{r = 1}^{6}^{56 - r} C_{3}

Q. The value of

^{50} C_{4} + 6 \sum r = 1

^{56 - r} C_{3}

Q. The value of

lim n \to \infty n \sum r = 1 \frac{r^{3}}{r^{4} + n^{4}}

(p - q)^{3} + (q - r)^{3} + (r - p)^{3}

Q. The value of

lim n \to \infty r = 4 n \sum r = 1 \frac{\sqrt{n}}{\sqrt{r} {(3 \sqrt{r} + \sqrt{n})}^{2}}

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Combinations

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon