Answer
Verified
392.4k+ views
Hint: Given in the question that selection must be made with at least one fruit among the given fruits. To find the probability we must find the probability of selecting 1 fruit, 2 fruits, 3 fruits and so on and add the probabilities to get the final probability.
Complete step-by-step answer:
Given, at least one fruit will be selected from $3$ bananas, $4$ apples and $2$ orange.
Total number of fruits $ = 3 + 4 + 2 = 9$
Hence, at least one fruit will be selected from the total $9$ number of fruits.
Therefore, Number of selection of at least one fruit = Number of selection of $1$ fruit + Number of selection of $2$ fruits + Number of selection of $3$ fruits +…………….+ Number of selection of $9$ fruits
$ = {}^9{C_1} + {}^9{C_2} + {}^9{C_3} + {}^9{C_4} + {}^9{C_5} + {}^9{C_6} + {}^9{C_7} + {}^9{C_8} + {}^9{C_9}$
We know that, ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
$ = \dfrac{{9!}}{{1!\left( {9 - 1} \right)!}} + \dfrac{{9!}}{{2!\left( {9 - 2} \right)!}} + \dfrac{{9!}}{{3!\left( {9 - 3} \right)!}} + \dfrac{{9!}}{{4!\left( {9 - 4} \right)!}} + \dfrac{{9!}}{{5!\left( {9 - 5} \right)!}} + \dfrac{{9!}}{{6!\left( {9 - 6} \right)!}} + \dfrac{{9!}}{{7!\left( {9 - 7} \right)!}} + \dfrac{{9!}}{{8!\left( {9 - 8} \right)!}} + \dfrac{{9!}}{{9!\left( {9 - 9} \right)!}}$
$ = \dfrac{{9!}}{{1!8!}} + \dfrac{{9!}}{{2!7!}} + \dfrac{{9!}}{{3!6!}} + \dfrac{{9!}}{{4!5!}} + \dfrac{{9!}}{{5!4!}} + \dfrac{{9!}}{{6!3!}} + \dfrac{{9!}}{{7!2!}} + \dfrac{{9!}}{{8!1!}} + \dfrac{{9!}}{{9!0!}}$
$ = \dfrac{{9 \times 8!}}{{1 \times 8!}} + \dfrac{{9 \times 8 \times 7!}}{{2 \times 1 \times 7!}} + \dfrac{{9 \times 8 \times 7 \times 6!}}{{3 \times 2 \times 1 \times 6!}} + \dfrac{{9 \times 8 \times 7 \times 6 \times 5!}}{{4 \times 3 \times 2 \times 1 \times 5!}} + \dfrac{{9 \times 8 \times 7 \times 6 \times 5!}}{{5! \times 4 \times 3 \times 2 \times 1}} + \dfrac{{9 \times 8 \times 7 \times 6!}}{{6! \times 3 \times 2 \times 1}} + \dfrac{{9 \times 8 \times 7!}}{{7! \times 2 \times 1}} + \dfrac{{9 \times 8!}}{{8! \times 1}} + \dfrac{{9!}}{{9!}}$
$ = 9 + 36 + 84 + 126 + 126 + 84 + 36 + 9 + 1$
$ = 511$
Thus, there are $511$ ways of selection of at least one fruit which can be made from $3$ bananas, $4$ apples and $2$ oranges.
Hence, option (B) is the correct answer.
Note: The combination is a selection of a part of a set of objects or selection of all objects when the order doesn’t matter. Therefore, the number of combinations of $n$ objects taken $r$ at a time is given by the formula: ${}^n{C_r} = \dfrac{{n\left( {n - 1} \right)\left( {n - 2} \right)........\left( {n - r + 1} \right)}}{{r!}} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Where the symbol denotes the factorial which means that the product of all the integers less than or equal to $n$ but it should be greater than or equal to $1$.
For example,
$0! = 1$
$1! = 1$
$2! = 2 \times 1 = 2$
$3! = 3 \times 2 \times 1 = 6$
Complete step-by-step answer:
Given, at least one fruit will be selected from $3$ bananas, $4$ apples and $2$ orange.
Total number of fruits $ = 3 + 4 + 2 = 9$
Hence, at least one fruit will be selected from the total $9$ number of fruits.
Therefore, Number of selection of at least one fruit = Number of selection of $1$ fruit + Number of selection of $2$ fruits + Number of selection of $3$ fruits +…………….+ Number of selection of $9$ fruits
$ = {}^9{C_1} + {}^9{C_2} + {}^9{C_3} + {}^9{C_4} + {}^9{C_5} + {}^9{C_6} + {}^9{C_7} + {}^9{C_8} + {}^9{C_9}$
We know that, ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
$ = \dfrac{{9!}}{{1!\left( {9 - 1} \right)!}} + \dfrac{{9!}}{{2!\left( {9 - 2} \right)!}} + \dfrac{{9!}}{{3!\left( {9 - 3} \right)!}} + \dfrac{{9!}}{{4!\left( {9 - 4} \right)!}} + \dfrac{{9!}}{{5!\left( {9 - 5} \right)!}} + \dfrac{{9!}}{{6!\left( {9 - 6} \right)!}} + \dfrac{{9!}}{{7!\left( {9 - 7} \right)!}} + \dfrac{{9!}}{{8!\left( {9 - 8} \right)!}} + \dfrac{{9!}}{{9!\left( {9 - 9} \right)!}}$
$ = \dfrac{{9!}}{{1!8!}} + \dfrac{{9!}}{{2!7!}} + \dfrac{{9!}}{{3!6!}} + \dfrac{{9!}}{{4!5!}} + \dfrac{{9!}}{{5!4!}} + \dfrac{{9!}}{{6!3!}} + \dfrac{{9!}}{{7!2!}} + \dfrac{{9!}}{{8!1!}} + \dfrac{{9!}}{{9!0!}}$
$ = \dfrac{{9 \times 8!}}{{1 \times 8!}} + \dfrac{{9 \times 8 \times 7!}}{{2 \times 1 \times 7!}} + \dfrac{{9 \times 8 \times 7 \times 6!}}{{3 \times 2 \times 1 \times 6!}} + \dfrac{{9 \times 8 \times 7 \times 6 \times 5!}}{{4 \times 3 \times 2 \times 1 \times 5!}} + \dfrac{{9 \times 8 \times 7 \times 6 \times 5!}}{{5! \times 4 \times 3 \times 2 \times 1}} + \dfrac{{9 \times 8 \times 7 \times 6!}}{{6! \times 3 \times 2 \times 1}} + \dfrac{{9 \times 8 \times 7!}}{{7! \times 2 \times 1}} + \dfrac{{9 \times 8!}}{{8! \times 1}} + \dfrac{{9!}}{{9!}}$
$ = 9 + 36 + 84 + 126 + 126 + 84 + 36 + 9 + 1$
$ = 511$
Thus, there are $511$ ways of selection of at least one fruit which can be made from $3$ bananas, $4$ apples and $2$ oranges.
Hence, option (B) is the correct answer.
Note: The combination is a selection of a part of a set of objects or selection of all objects when the order doesn’t matter. Therefore, the number of combinations of $n$ objects taken $r$ at a time is given by the formula: ${}^n{C_r} = \dfrac{{n\left( {n - 1} \right)\left( {n - 2} \right)........\left( {n - r + 1} \right)}}{{r!}} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Where the symbol denotes the factorial which means that the product of all the integers less than or equal to $n$ but it should be greater than or equal to $1$.
For example,
$0! = 1$
$1! = 1$
$2! = 2 \times 1 = 2$
$3! = 3 \times 2 \times 1 = 6$
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Draw a diagram showing the external features of fish class 11 biology CBSE
Select the word that is correctly spelled a Twelveth class 10 english CBSE
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
What is BLO What is the full form of BLO class 8 social science CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Change the following sentences into negative and interrogative class 10 english CBSE
Fill the blanks with proper collective nouns 1 A of class 10 english CBSE