Let S denote the 10-element set {a,b,c,d,e,f,g,h,i,j}. How many ways can we cons
ID: 3071633 • Letter: L
Question
Let S denote the 10-element set {a,b,c,d,e,f,g,h,i,j}. How many ways can we construct a subset of S of size 7 ? 120 How many ways can we construct a subset of S of size 7 containing the element j? 84 How many ways can we construct a subset of S of size 7 containing i but not j ? 28 How many ways can we construct a subset of S of size 7 containing h but neither i nor j ? 7 How many ways can we construct a subset of S of size 7 containing g but not h, i or j? 1 Note that every subset of S of size 7 falls into exactly one of the categories described in parts (b) through (e) above. Use that fact to derive a summation formula involving expressions nCr.
Explanation / Answer
(a) How many ways can we construct a subset of S of size 7 ? 120
answer is 10C7=10!/(7!*(10-7)!)=10!/(7!*3!)=120
(b) How many ways can we construct a subset of S of size 7 containing the element j? 84
since j is already selected, now we have to select 6 from 9 element set( {a,b,c,d,e,f,g,h,i}
answer will be 9C6= (9!)/(6!*3!)= 84
(c) How many ways can we construct a subset of S of size 7 containing i but not j ? 28
here we donot want j, it means now we have 9 element {a,b,c,d,e,f,g,h,i}
also we want i to be selected so we we now select 6 element from 8 element set {a,b,c,d,e,f,g,h}
answer will be 8C6=8!/(6!*2!)=28
(d) How many ways can we construct a subset of S of size 7 containing h but neither i nor j ? 7
here now we have effective set {a,b,c,d,e,f,g,h} and want to select h compulsory so
answer will be 7C6=7
(e) here effective set is {a,b,c,d,e,f,g}
g to be selected cumpulsory so number of way will be here 6C6=1
How many ways can we construct a subset of S of size 7 containing g but not h, i or j? 1 Note that every subset of S of size 7 falls into exactly one of the categories described in parts (b) through (e) above. Use that fact to derive a summation formula involving expressions nCr.
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