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Question

The value of 50C4 + not stretchy sum subscript r equals 1 end subscript superscript 6 end superscript 56 minus r C subscript 3 end subscriptis -

  1. 55C4    
  2. 55C3    
  3. 56C3    
  4. 56C4    

hintHint:

By choosing some items from a set and creating subsets, permutation and combination are two approaches to represent a group of objects. It outlines the numerous configurations for a particular set of data. Permutations are the selection of data or objects from a set, whereas combinations are the order in which they are represented. Here we have to find the value of 50C4 + not stretchy sum subscript r equals 1 end subscript superscript 6 end superscript 56 minus r C subscript 3 end subscript.

The correct answer is: 56C4


    A permutation is the non-replaceable selection of r items from a set of n items in which the order is important.
    P presuperscript n subscript r equals space fraction numerator left parenthesis n factorial right parenthesis space over denominator left parenthesis n minus r right parenthesis factorial end fraction
    A combination is created by selecting r items from a group of n items without replacing them and without regard to their order.
    C presuperscript n subscript r equals space fraction numerator left parenthesis n factorial right parenthesis space over denominator left parenthesis n minus r right parenthesis factorial cross times r factorial end fraction
    Here we have given:
    C presuperscript 50 subscript 4 plus begin inline style sum from r equals 1 to 6 of end style space C presuperscript 56 minus r end presuperscript subscript 3
S o l v i n g space t h i s comma space w e space g e t colon
C presuperscript 50 subscript 4 plus C presuperscript 50 subscript 3 plus C presuperscript 51 subscript 3 plus C presuperscript 52 subscript 3 plus C presuperscript 53 subscript 3 plus C presuperscript 54 subscript 3 plus C presuperscript 55 subscript 3
N o w space u s n i n g space t h e space f o r m u l a colon
C presuperscript n subscript i plus C presuperscript n subscript i minus 1 end subscript equals C presuperscript n plus 1 end presuperscript subscript i
A p p l y i n g space t h i s comma space w e space g e t colon
C presuperscript 51 subscript 4 plus C presuperscript 51 subscript 3 plus C presuperscript 52 subscript 3 plus C presuperscript 53 subscript 3 plus C presuperscript 54 subscript 3 plus C presuperscript 55 subscript 3
A p p l y i n g space t h e space s a m e space f o r m u l a space a g a i n comma space w e space g e t colon
C presuperscript 52 subscript 4 plus C presuperscript 52 subscript 3 plus C presuperscript 53 subscript 3 plus C presuperscript 54 subscript 3 plus C presuperscript 55 subscript 3
A p p l y i n g space t h e space s a m e space f o r m u l a space a g a i n comma space w e space g e t colon
C presuperscript 53 subscript 4 plus C presuperscript 53 subscript 3 plus C presuperscript 54 subscript 3 plus C presuperscript 55 subscript 3
A p p l y i n g space t h e space s a m e space f o r m u l a space a g a i n comma space w e space g e t colon
C presuperscript 54 subscript 4 plus C presuperscript 54 subscript 3 plus C presuperscript 55 subscript 3
A p p l y i n g space t h e space s a m e space f o r m u l a space a g a i n comma space w e space g e t colon
C presuperscript 55 subscript 4 plus C presuperscript 55 subscript 3
A p p l y i n g space t h e space s a m e space f o r m u l a space a g a i n comma space w e space g e t colon
C presuperscript 56 subscript 4
     

    The different ways in which items from a set may be chosen, usually without replacement, to construct subsets, are called permutations and combinations. When the order of the selection is a consideration, this selection of subsets is referred to as a permutation; when it is not, it is referred to as a combination. So the final answer is C presuperscript 56 subscript 4.

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