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Maths NCERT Exemplar Solutions Class 11th Chapter Seven

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Maths NCERT Exemplar Solutions Class 11th Chapter Seven Exemplar Solution

There will be only 24 selections containing at least one red ball out of a bag containing 4 red and 5 black balls. It is given that the balls of the same colour are identical.

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Payal Gupta

Contributor-Level 10

This is a True or False Type Questions as classified in NCERT Exemplar

$\begin{array}{l} W e h a v e 4 r e d a n d 5 b l a c k b a l l s i n a b o x a n d a t l e a s t o n e r e d b a l l i s t o b e d r a w n \\ ∴ N u m b e r o f s e l e c t i o n = [(4 + 1) (5 + 1) - 1] - 5 = [5 * 6 - 1] - 5 \\ 29 - 5 = 24 \\ H e n c e, t h e g i v e n s t a t e m e n t i s' T r u e' . \end{array}$

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Maths NCERT Exemplar Solutions Class 11th Chapter Seven Exemplar Solution

If some or all of $n$ objects are taken at a time, the number of combinations is $2^{n} - 1$ .

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Payal Gupta

Contributor-Level 10

This is a True or False Type Questions as classified in NCERT Exemplar

$\begin{array}{l} W h e n s o m e o r a l l o b j e c t s, t a k e n a t a t i m e, t h e n t h e n u m b e r o f s e l e c t i o n w i l l b e \\ {}_{n}{C_{1}} + {}_{n}{C_{2}} + {}_{n}{C_{3}} + \dots + {}_{n}{C_{n}} = 2^{n} - 1 [? {}_{n}{C_{0}} + {}_{n}{C_{1}} + {}_{n}{C_{2}} + {}_{n}{C_{3}} + \dots + {}_{n}{C_{n}} = 2^{n}] \\ H e n c e, t h e g i v e n s t a t e m e n t i s' T r u e' . \end{array}$

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Maths NCERT Exemplar Solutions Class 11th Chapter Seven Exemplar Solution

In a steamer, there are stalls for 12 animals, and there are horses, cows, and calves (not less than 12 each) ready to be shipped. They can be loaded in 312 ways.

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Payal Gupta

Contributor-Level 10

This is a True or False Type Questions as classified in NCERT Exemplar

$\begin{array}{l} T h e r e a r e 3 t y p e s o f a n i m a l s h o r s e s, c o w s a n d c a l v e s n o t l e s s t h a n 12 e a c h . \\ S o, n u m b e r o f w a y s o f l o a d i n g = 3^{12} \\ H e n c e, t h e g i v e n s t a t e m e n t i s' T r u e' . \end{array}$

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Maths NCERT Exemplar Solutions Class 11th Chapter Seven Exemplar Solution

In the permutations of $n$ things, $r$ taken together, the number of permutations in which $m$ particular things occur together is $n - m P_{r} - m * r P m$ .

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Payal Gupta

Contributor-Level 10

This is a True or False Type Questions as classified in NCERT Exemplar

$\begin{array}{l} A r r a n g e m e n t o f n t h i n g s, r t a k e n a t a t i m e i n w h i c h m t h i n g s o c c u r t o g e t h e r . \\ S o, n u m b e r o f o b j e c t e x c l u d i n g m o b j e c t = (r - m) \\ H e r e, w e f i r s t a r r a n g e (r - m + 1) o b j e c t \\ ∴ N u m b e r o f a r r a n g e m e n t s = (r - m + 1)! \\ m o b j e c t s c a n b e a r r a n g e d i n m! w a y s \\ S o, t h e r e q u i r e d n u m b e r o f a r r a n g e m e n t s = (r - m + 1)! * m! \\ H e n c e, t h e g i v e n s t a t e m e n t i s' F a l s e' . \end{array}$

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New answer posted

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Maths NCERT Exemplar Solutions Class 11th Chapter Seven Exemplar Solution

Three letters can be posted in five letterboxes in 35 ways.

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Payal Gupta

Contributor-Level 10

This is a True or False Type Questions as classified in NCERT Exemplar

$\begin{array}{l} G i v e n t h a t 3 l e t t e r s a r e t o b e p o s t e d i n 5 l e t t e r b o x e s \\ ∴ Required n u m b e r o f w a y s = 5^{3} = 125 \\ H e n c e, t h e g i v e n s t a t e m e n t i s' F a l s e' \end{array}$

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Maths NCERT Exemplar Solutions Class 11th Chapter Seven Exemplar Solution

There are 12 points in a plane of which 5 points are collinear, then the number of lines obtained by joining these points in pairs is ¹²C₂ - ⁵C₂.

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Payal Gupta

Contributor-Level 10

This is a True or False Type Questions as classified in NCERT Exemplar

$\begin{array}{l} Required n u m b e r o f l i n e s = {}_{12}{C_{2}} - {}_{5}{C_{2}} + 1 \\ H e n c e, t h e g i v e n s t a t e m e n t i s' F a l s e' \end{array}$

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Maths NCERT Exemplar Solutions Class 11th Chapter Seven Exemplar Solution

A box contains 2 white balls, 3 black balls, and 4 red balls. The number of ways three balls can be drawn from the box if at least one black ball is to be included in the draw is ______.

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Payal Gupta

Contributor-Level 10

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar

$\begin{array}{l} W e h a v e 2 w h i t e, 3 b l a c k a n d 4 r e d b a l l s \\ I t i s g i v e n t h a t a t l e a s t 1 b l a c k b a l l i s t o b e i n c l u d e d . \\ ∴ Required n u m b e r o f w a y s = {}_{3}{C_{1}} * {}_{6}{C_{2}} + {}_{3}{C_{2}} * {}_{6}{C_{1}} + {}_{3}{C_{3}} \\ = 3 * 15 + 3 * 6 + 1 = 45 + 18 + 1 = 64 \\ H e n c e, t h e v a l u e o f t h e f i l l e r i s 64 . \end{array}$

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New answer posted

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Maths NCERT Exemplar Solutions Class 11th Chapter Seven Exemplar Solution

A committee of 6 is to be chosen from 10 men and 7 women so as to contain at least 3 men and 2 women. In how many different ways can this be done if two particular women refuse to serve on the same committee?
[Hint: At least 3 men and 2 women: The number of ways = ¹⁰C₃ * ¹⁰C₃ + ¹⁰C₄ * ⁷C₂. For 2 particular women to be always there: the number of ways = ¹⁰C₄ * ¹⁰C₃ * ⁵C₁. The total number of committees when two particular women are never together = Total – together.]

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Payal Gupta

Contributor-Level 10

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar

$\begin{array}{l} W e h a v e 10 m e n a n d 7 w o m e n o u t o f w h i c h a c o m m i t t e e o f 6 i s t o b e f o r m e d w h i c h c o n t a i n \\ a t l e a s t 3 m e n a n d 2 w o m e n \\ T h e r e f o r e, N u m b e r o f w a y s = {}_{10}{C_{3}} * {}_{7}{C_{3}} + {}_{10}{C_{4}} * {}_{7}{C_{2}} \\ = \frac{10 * 9 * 8}{3 * 2 * 1} * \frac{7 * 6 * 5}{3 * 2 * 1} + \frac{10 * 9 * 8 * 7}{4 * 3 * 2 * 1} * \frac{7 * 6}{2 * 1} \\ = 120 * 35 + 210 * 21 = 4200 + 4410 = 8610 \\ I f 2 p a r t i c u l a r w o m e n t o b e a l w a y s p r e s e n t, t h e n t h e n u m b e r o f w a y s = {}_{10}{C_{4}} * {}_{5}{C_{0}} + {}_{10}{C_{3}} * {}_{5}{C_{1}} \\ = \frac{10 * 9 * 8 * 7}{4 * 3 * 2 * 1} * 1 + \frac{10 * 9 * 8}{3 * 2 * 1} * 5 = 210 + 120 * 5 \\ = 210 + 600 = 810 \\ ∴ T o t a l n u m b e r o f c o m m i t t e e = 8610 - 810 = 7800 \\ H e n c e, t h e v a l u e o f t h e f i l l e r i s 7800 . \end{array}$

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Maths NCERT Exemplar Solutions Class 11th Chapter Seven Exemplar Solution

The total number of ways in which six '+' and four '–' signs can be arranged in a line such that no two signs '–' occur together is ______.

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Payal Gupta

Contributor-Level 10

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar

$\begin{array}{l} T h e f o l l o w i n g m a y b e t h e a r r a n g e m e n t o f (-) a n d (+) \\ (-) (+) (-) (+) (-) (+) (-) (+) (-) (+) (-) (+) (-) \\ T h e r e f o r e,' +' s i g n c a n b e a r r a n g e d o n l y i s 1 w a y b e c a u s e a l l a r e i d e n t i c a l . \\ a n d 4 (-) s i g n s c a n b e a r r a n g e d a t 7 p l a c e s i n {}_{7}{C_{4}} w a y s \\ ∴ T o t a l n u m b e r o f w a y s = {}_{7}{C_{4}} * 1 = \frac{7 * 6 * 5 * 4}{4 * 3 * 2 * 1} * 1 = 35 w a y s \\ H e n c e, t h e v a l u e o f t h e f i l l e r i s 35 . \end{array}$

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New answer posted

3 months ago

Maths NCERT Exemplar Solutions Class 11th Chapter Seven Exemplar Solution

In a football championship, 153 matches were played. Every two teams played one match with each other. The number of teams participating in the championship is ______.

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Payal Gupta

Contributor-Level 10

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar

$\begin{array}{l} L e t t h e n u m b e r o f p a r t i c i p a t i n g t e a m s b e n . \\ G i v e n t h a t e v e r y t w o t e a m s p l a y e d o n e m a t c h w i t h e a c h o t h e r . \\ ∴ T o t a l n u m b e r o f m a t c h e s p l a y e d = {}_{n}{C_{2}} \\ S o {}_{n}{C_{2}} = 153 \\ \Rightarrow \frac{n (n - 1)}{2} = 153 \Rightarrow n^{2} - n = 306 \\ \Rightarrow n^{2} - n - 306 = 0 \Rightarrow n^{2} - 18 n + 17 n - 306 = 0 \\ \Rightarrow n (n - 18) + 17 (n - 18) = 0 \Rightarrow (n - 18) (n + 17) = 0 \\ \Rightarrow n - 18 = 0 a n d n + 17 = 0 \Rightarrow n = 18, n \neq - 17 \\ H e n c e, t h e v a l u e o f t h e f i l l e r i s 18 . \end{array}$

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